Unit 9: Polynomials & Factoring

Unit 9: Polynomials & Factoring Warm-Up Write a personal definition for what you think a “factor” is. Review Let’s review some vocabulary. . . . . . . Coefficient – The number in front of the variable. Constant – A number without a variable. Terms – Any number, variable, or product or quotient of a number and variable. (always separated by + or – signs) Polynomial These are examples of polynomials: 5X² - 2X + 3 – 3X³ + 7X - 4 Degree of a Polynomial The degree of a polynomial is calculated by finding the largest exponent in the polynomial. Degree of a Polynomial 1 2x Degree of a Polynomial 2x Largest exponent is 1 Degree of a Polynomial 2x 2 3x − 5 Largest exponent is 1 degree 1 Degree of a Polynomial 2x Largest exponent is 1 2 Largest exponent is 2 3x − 5 degree 1 Degree of a Polynomial 2x Largest exponent is 1 degree 1 2 Largest exponent is 2 degree 2 3x − 5 3 5y + 2y Degree of a Polynomial 2x Largest exponent is 1 degree 1 2 Largest exponent is 2 degree 2 3x − 5 3 5y + 2y Largest exponent is 3 Degree of a Polynomial 2x Largest exponent is 1 degree 1 2 Largest exponent is 2 degree 2 Largest exponent is 3 degree 3 3x − 5 3 5y + 2y 7x n Degree of a Polynomial 2x Largest exponent is 1 degree 1 2 Largest exponent is 2 degree 2 Largest exponent is 3 degree 3 3x − 5 3 5y + 2y 7x n Largest exponent is n Degree of a Polynomial 2x Largest exponent is 1 degree 1 2 Largest exponent is 2 degree 2 Largest exponent is 3 degree 3 Largest exponent is n degree n 3x − 5 3 5y + 2y 7x n Example 1 Answer the following questions about this polynomial: 6 3 5a + 2a − 7 a + 11 1. What is the degree of the polynomial? 6 Example 1 Answer the following questions about this polynomial: 6 3 5a + 2a − 7 a + 11 1. What is the degree of the polynomial? 6 2. What is the constant? Example 1 Answer the following questions about this polynomial: 5a 6 + 2a 3 − 7 a + 11 1. What is the degree of the polynomial? 2. What is the constant? 11 6 Example 1 Answer the following questions about this polynomial: 6 3 5a + 2a − 7 a + 11 1. What is the degree of the polynomial? 2. What is the constant? 11 6 3. What is coefficient of third degree term? 2 Example 2 Answer the following questions about this polynomial: 4 2 9 x + 17 x − 5 1. What is the degree of the polynomial? 2. What is the constant? –5 4 3. What is coefficient of second degree term? 17 Standard Form of Polynomials When you write polynomials, you write them with the exponents in decreasing order. 7 6 5 4 3 2 4 x + 10 x − 3x + 12 x − 2 x − 5 x − 8 x + 6 Example 3 Write this polynomial in standard form. 2 4 5 3 5x + 8x − 6 x + 9 x + 2 x − 4 What term has the largest exponent? −6x 5 Example 3 Write this polynomial in standard form. 2 4 5 3 5x + 8x − 6 x + 9 x + 2 x − 4 What term has the next largest exponent? 5 −6x +8x 4 Example 3 Write this polynomial in standard form. 2 4 5 3 5x + 8x − 6 x + 9 x + 2 x − 4 What term has the next largest exponent? 4 −6x +8x +9x 5 3 Example 3 Write this polynomial in standard form. 2 4 5 3 5x + 8x − 6 x + 9 x + 2 x − 4 What term has the next largest exponent? 4 3 −6x +8x +9x +5x 5 2 Example 3 Write this polynomial in standard form. 2 4 5 3 5x + 8x − 6 x + 9 x + 2 x − 4 What term has the next largest exponent? −6x +8x +9x +5x +2x 5 4 3 2 Example 3 Write this polynomial in standard form. 2 4 5 3 5x + 8x − 6 x + 9 x + 2 x − 4 What is the last term? −6x +8x +9x +5x +2x −4 5 4 3 2 TRY THESE: Copy them onto notebook paper first, then re-write in standard form. SOLUTIONS: 1. 4x – 1. -3X2 + 4X 3x2 2. 4X + 9 (stay same) 2. 4x + 9 3. 6 – 3x – 7x2 3. -7x2 – 3x + 6 3 + 9z2 + 5z – 5 4. -11z 4. – 5z – 5 8 – 7y5 + y 5 8 5. 15y 5. y – 7y + 15y 2 + 4c - 2 2 6. c 6. c – 2 + 4c 9z2 11z3 + continued. . . SOLUTIONS: 7. 4q4 + 3q2 – 8q – 10 7. -10 + 4q4 – 8q + 3q2 8. 7x – 2 8. -2 + 7x 9. -x4 + 2x3 + 3 9. 2x3 – x4 + 3 10. 3x4 – 2x3 + 7x – 2 10. 3x4 – 2 – 2x3 + 7x 11. -2x + 5 11. 5 – 2x 12. 3x6 + 2x2 – 4 12. 3x6 – 4 + 2x2 13. x3 – 4x2 – 2x + 5 13. -2x + 5 – 4x2 + x3 INDEPENDENT PRACTICE (classwork or homework – follow teacher instructions) Copy and re-write in standard form. 1. 6x2 + 7 – 9x4 7. 7z3 – 12 + 4z 2. 3y – 9 – y3 8. 3 – 3x – 4x2 – 6x3 3. 9 + 7v 9. 1 + 5x + 2x2 4. 2x + 5. 4x2 10.5y + 18y8 – 3y2 7x2 – 2x – 6. -2x + 3 + x3 –4 5x2 11.-x – 4 – 3x2 - 6x3 12.9y – 9y3 + 9y2 Degree of a Polynomial The degree of a polynomial is calculated by finding the largest exponent in the polynomial. (Learned in previous lesson on Writing Polynomials in Standard Form) Degree of a Polynomial (Each degree has a special “name”) 9 Degree of a Polynomial (Each degree has a special “name”) 9 No variable Degree of a Polynomial (Each degree has a special “name”) 9 No variable Constant Degree of a Polynomial (Each degree has a special “name”) 9 8x No variable Constant Degree of a Polynomial (Each degree has a special “name”) 9 8x No variable 1st degree Constant Degree of a Polynomial (Each degree has a special “name”) 9 8x No variable Constant 1st degree Linear Degree of a Polynomial (Each degree has a special “name”) 9 8x 7x2 + 3x No variable Constant 1st degree Linear Degree of a Polynomial (Each degree has a special “name”) 9 8x 7x2 + 3x No variable Constant 1st degree Linear 2nd degree Degree of a Polynomial (Each degree has a special “name”) 9 8x 7x2 + 3x No variable Constant 1st degree Linear 2nd degree Quadratic Degree of a Polynomial (Each degree has a special “name”) 9 8x 7x2 + 3x 6x3 – 2x No variable Constant 1st degree Linear 2nd degree Quadratic Degree of a Polynomial (Each degree has a special “name”) 9 8x 7x2 + 3x 6x3 – 2x No variable Constant 1st degree Linear 2nd degree Quadratic 3rd degree Degree of a Polynomial (Each degree has a special “name”) 9 8x 7x2 + 3x 6x3 – 2x No variable Constant 1st degree Linear 2nd degree Quadratic 3rd degree Cubic Degree of a Polynomial (Each degree has a special “name”) 9 8x 7x2 + 3x 6x3 – 2x 3x4 + 5x – 1 No variable Constant 1st degree Linear 2nd degree Quadratic 3rd degree Cubic Degree of a Polynomial (Each degree has a special “name”) 9 8x 7x2 + 3x 6x3 – 2x 3x4 + 5x – 1 No variable Constant 1st degree Linear 2nd degree Quadratic 3rd degree Cubic 4th degree Degree of a Polynomial (Each degree has a special “name”) 9 8x 7x2 + 3x 6x3 – 2x 3x4 + 5x – 1 No variable Constant 1st degree Linear 2nd degree Quadratic 3rd degree Cubic 4th degree Quartic Degree of a Polynomial (Each degree has a special “name”) 9 8x 7x2 + 3x 6x3 – 2x 3x4 + 5x – 1 2x5 + 7x3 No variable Constant 1st degree Linear 2nd degree Quadratic 3rd degree Cubic 4th degree Quartic Degree of a Polynomial (Each degree has a special “name”) 9 8x 7x2 + 3x 6x3 – 2x 3x4 + 5x – 1 2x5 + 7x3 No variable Constant 1st degree Linear 2nd degree Quadratic 3rd degree Cubic 4th degree Quartic 5th degree Degree of a Polynomial (Each degree has a special “name”) 9 8x 7x2 + 3x 6x3 – 2x 3x4 + 5x – 1 2x5 + 7x3 No variable Constant 1st degree Linear 2nd degree Quadratic 3rd degree Cubic 4th degree Quartic 5th degree Quintic Degree of a Polynomial (Each degree has a special “name”) 9 8x 7x2 + 3x 6x3 – 2x 3x4 + 5x – 1 2x5 + 7x3 5xn No variable Constant 1st degree Linear 2nd degree Quadratic 3rd degree Cubic 4th degree Quartic 5th degree Quintic Degree of a Polynomial (Each degree has a special “name”) 9 8x 7x2 + 3x 6x3 – 2x 3x4 + 5x – 1 2x5 + 7x3 5xn No variable Constant 1st degree Linear 2nd degree Quadratic 3rd degree Cubic 4th degree Quartic 5th degree Quintic “nth” degree Degree of a Polynomial (Each degree has a special “name”) 9 8x 7x2 + 3x 6x3 – 2x 3x4 + 5x – 1 2x5 + 7x3 5xn No variable Constant 1st degree Linear 2nd degree Quadratic 3rd degree Cubic 4th degree Quartic 5th degree Quintic “nth” degree “nth” degree DEGREE Let’s practice classifying polynomials by NAME “degree”. POLYNOMIAL 1. 3z4 + 5z3 – 7 2. 15a + 25 3. 185 4. 2c10 – 7c6 + 4c3 - 9 5. 2f3 – 7f2 + 1 6. 15y2 7. 9g4 – 3g + 5 8. 10r5 –7r 9. 16n7 + 6n4 – 3n2 1. Quartic 2. Linear 3. Constant 4. Tenth degree 5. Cubic 6. Quadratic 7. Quartic 8. Quintic 9. Seventh degree The degree name becomes the “first name” of the polynomial. Naming Polynomials (by number of terms) 5x 2 Naming Polynomials (by number of terms) 5x 2 One term Naming Polynomials (by number of terms) 5x 2 One term Monomial Naming Polynomials (by number of terms) 5x 2 x+3 One term Monomial Naming Polynomials (by number of terms) 5x 2 x+3 One term Two terms Monomial Naming Polynomials (by number of terms) 5x 2 x+3 One term Monomial Two terms Binomial Naming Polynomials (by number of terms) 5x 2 x+3 7 x3 + 2 x − 4 One term Monomial Two terms Binomial Naming Polynomials (by number of terms) 5x 2 One term Monomial x+3 Two terms Binomial 7 x3 + 2 x − 4 Three terms Naming Polynomials (by number of terms) 5x 2 One term Monomial x+3 Two terms Binomial 7 x3 + 2 x − 4 Three terms Trinomial Naming Polynomials (by number of terms) 2 One term Monomial x+3 Two terms Binomial Three terms Trinomial 5x 3 7x + 2x − 4 x 4 + 2 x3 − 2 x 2 + 5 x Naming Polynomials (by number of terms) 5x 2 One term Monomial Two terms Binomial 7x + 2x − 4 Three terms Trinomial x 4 + 2 x3 − 2 x 2 + 5 x Four (or more) terms x+3 3 Naming Polynomials (by number of terms) 5x 2 One term Monomial x+3 Two terms Binomial 7 x3 + 2 x − 4 Three terms Trinomial 4 3 2 x + 2 x − 2 x + 5x Four (or more) Polynomial with 4 (or more) terms terms Classifying (or Naming) a Polynomial by Number of Terms Classifying (or Naming) a Polynomial by Number of Terms 25 1 term Classifying (or Naming) a Polynomial by Number of Terms 25 1 term Monomial Classifying (or Naming) a Polynomial by Number of Terms 25 15h 1 term Monomial Classifying (or Naming) a Polynomial by Number of Terms 25 1 term 15h 1 term Monomial Classifying (or Naming) a Polynomial by Number of Terms 25 1 term Monomial 15h 1 term Monomial Classifying (or Naming) a Polynomial by Number of Terms 25 1 term Monomial 15h 1 term Monomial 25h3 + 7 Classifying (or Naming) a Polynomial by Number of Terms 25 1 term Monomial 15h 1 term Monomial 25h3 + 7 2 terms Classifying (or Naming) a Polynomial by Number of Terms 25 1 term Monomial 15h 1 term Monomial 25h3 + 7 2 terms Binomial Classifying (or Naming) a Polynomial by Number of Terms 25 1 term Monomial 15h 1 term Monomial 25h3 + 7 2 terms Binomial 3t7 – 4t6 + 12 Classifying (or Naming) a Polynomial by Number of Terms 25 1 term Monomial 15h 1 term Monomial 25h3 + 7 2 terms Binomial 3t7 – 4t6 + 12 3 terms Classifying (or Naming) a Polynomial by Number of Terms 25 1 term Monomial 15h 1 term Monomial 25h3 + 7 2 terms Binomial 3t7 – 4t6 + 12 3 terms Trinomial Classifying (or Naming) a Polynomial by Number of Terms 25 1 term Monomial 15h 1 term Monomial 25h3 + 7 2 terms Binomial 3t7 – 4t6 + 12 3 terms Trinomial 4b10 + 2b8 – 7b6 - 8 Classifying (or Naming) a Polynomial by Number of Terms 25 1 term Monomial 15h 1 term Monomial 25h3 + 7 2 terms Binomial 3t7 – 4t6 + 12 3 terms Trinomial 4b10 + 2b8 – 7b6 - 8 4 (or more) terms Classifying (or Naming) a Polynomial by Number of Terms 25 1 term Monomial 15h 1 term Monomial 25h3 + 7 2 terms Binomial 3t7 – 4t6 + 12 3 terms Trinomial 4b10 + 2b8 – 7b6 - 8 4 (or more) terms Polynomial with 4 terms Classifying (or Naming) a Polynomial by Number of Terms 25 1 term Monomial 15h 1 term Monomial 25h3 + 7 2 terms Binomial 3t7 – 4t6 + 12 3 terms Trinomial 4b10 + 2b8 – 7b6 - 8 4 (or more) terms Polynomial with 4 terms 8j7 – 3j5 + 2j4 - j2 - 6j + 7 Classifying (or Naming) a Polynomial by Number of Terms 25 1 term Monomial 15h 1 term Monomial 25h3 + 7 2 terms Binomial 3t7 – 4t6 + 12 3 terms Trinomial 4b10 + 2b8 – 7b6 - 8 4 (or more) terms Polynomial with 4 terms 8j7 – 3j5 + 2j4 - j2 - 6j + 7 6 terms Classifying (or Naming) a Polynomial by Number of Terms 25 1 term Monomial 15h 1 term Monomial 25h3 + 7 2 terms Binomial 3t7 – 4t6 + 12 3 terms Trinomial 4b10 + 2b8 – 7b6 - 8 4 (or more) terms Polynomial with 4 terms 8j7 – 3j5 + 2j4 - j2 - 6j + 7 6 terms Polynomial with 6 terms The term name becomes the “last name” of a polynomial. Let’s practice classifying a polynomial by “number of terms”. Polynomial Classify by # of Terms: 1. Monomial 1. 15x 2. 2e8 3. 6c + 5 4. 3y7 5. 64 6. 2p8 – 4p6 + 9p4 + 3p – 1 6. Polynomial with 5 terms 7. 25h3 – 15h2 + 18 7. Trinomial 8. 55c19 + 35 8. Binomial – – 3e7 4y5 + 3e – 7 + 8y3 2. Polynomial with 4 terms 3. Binomial 4. Trinomial 5. Monomial Naming/Classifying Polynomials by Degree (1st name) and Number of Terms (last name) Polynomial 2 2x 2x + 7 5b2 Degree # of Terms Full Name Naming/Classifying Polynomials by Degree (1st name) and Number of Terms (last name) Polynomial Degree 2 No degree (constant) 2x 2x + 7 5b2 # of Terms Full Name Naming/Classifying Polynomials by Degree (1st name) and Number of Terms (last name) Polynomial Degree # of Terms 2 No degree (constant) 1 term (monomial) 2x 2x + 7 5b2 Full Name Naming/Classifying Polynomials by Degree (1st name) and Number of Terms (last name) Polynomial Degree 2 No degree (constant) 2x 2x + 7 5b2 # of Terms Full Name 1 term Constant (monomial) Monomial Naming/Classifying Polynomials by Degree (1st name) and Number of Terms (last name) Polynomial Degree 2 No degree (constant) 2x 1st degree (Linear) 2x + 7 5b2 # of Terms Full Name 1 term Constant (monomial) Monomial Naming/Classifying Polynomials by Degree (1st name) and Number of Terms (last name) Polynomial Degree 2 No degree (constant) 1 term Constant (monomial) Monomial 2x 1st degree (Linear) 1 term (monomial) 2x + 7 5b2 # of Terms Full Name Naming/Classifying Polynomials by Degree (1st name) and Number of Terms (last name) Polynomial Degree 2 No degree (constant) 1 term Constant (monomial) Monomial 2x 1st degree (Linear) 1 term Linear (monomial) Monomial 2x + 7 5b2 # of Terms Full Name Naming/Classifying Polynomials by Degree (1st name) and Number of Terms (last name) Polynomial Degree 2 No degree (constant) 1 term Constant (monomial) Monomial 2x 1st degree (Linear) 1 term Linear (monomial) Monomial 2x + 7 1st degree (linear) 5b2 # of Terms Full Name Naming/Classifying Polynomials by Degree (1st name) and Number of Terms (last name) Polynomial Degree 2 No degree (constant) 1 term Constant (monomial) Monomial 2x 1st degree (Linear) 1 term Linear (monomial) Monomial 2x + 7 1st degree (linear) 5b2 # of Terms 2 terms (binomial) Full Name Naming/Classifying Polynomials by Degree (1st name) and Number of Terms (last name) Polynomial Degree 2 No degree (constant) 1 term Constant (monomial) Monomial 2x 1st degree (Linear) 1 term Linear (monomial) Monomial 2x + 7 1st degree (linear) 5b2 # of Terms 2 terms (binomial) Full Name Linear Binomial Naming/Classifying Polynomials by Degree (1st name) and Number of Terms (last name) Polynomial Degree # of Terms 2 No degree (constant) 1 term Constant (monomial) Monomial 2x 1st degree (Linear) 1 term Linear (monomial) Monomial 2x + 7 1st degree (linear) 5b2 2nd degree (quadratic) 2 terms (binomial) Full Name Linear Binomial Naming/Classifying Polynomials by Degree (1st name) and Number of Terms (last name) Polynomial Degree # of Terms Full Name 2 No degree (constant) 1 term Constant (monomial) Monomial 2x 1st degree (Linear) 1 term Linear (monomial) Monomial 2x + 7 1st degree (linear) 2 terms (binomial) 5b2 2nd degree (quadratic) 1 term (monomial) Linear Binomial Naming/Classifying Polynomials by Degree (1st name) and Number of Terms (last name) Polynomial Degree # of Terms 2 No degree (constant) 1 term Constant (monomial) Monomial 2x 1st degree (Linear) 1 term Linear (monomial) Monomial 2x + 7 1st degree (linear) 5b2 2nd degree (quadratic) 2 terms (binomial) Full Name Linear Binomial 1 term Quadratic (monomial) Monomial . . .more polynomials. . . 5b2 + 2 5b2 + 3b + 2 3f3 2f3 – 7 7f3 – 2f2 + f 9k4 . . .more polynomials. . . 5b2 + 2 5b2 + 3b + 2 3f3 2f3 – 7 7f3 – 2f2 + f 9k4 2nd degree (quadratic) . . .more polynomials. . . 5b2 + 2 5b2 + 3b + 2 3f3 2f3 – 7 7f3 – 2f2 + f 9k4 2nd degree (quadratic) 2 terms (binomial) . . .more polynomials. . . 5b2 + 2 5b2 + 3b + 2 3f3 2f3 – 7 7f3 – 2f2 + f 9k4 2nd degree (quadratic) 2 terms Quadratic (binomial) Binomial . . .more polynomials. . . 5b2 + 2 5b2 + 3b + 2 3f3 2f3 – 7 7f3 – 2f2 + f 9k4 2nd degree (quadratic) 2nd degree (quadratic) 2 terms Quadratic (binomial) Binomial . . .more polynomials. . . 5b2 + 2 5b2 + 3b + 2 3f3 2f3 – 7 7f3 – 2f2 + f 9k4 2nd degree (quadratic) 2nd degree (quadratic) 2 terms Quadratic (binomial) Binomial 3 terms (trinomial) . . .more polynomials. . . 5b2 + 2 5b2 + 3b + 2 3f3 2f3 – 7 7f3 – 2f2 + f 9k4 2nd degree (quadratic) 2nd degree (quadratic) 2 terms (binomial) 3 terms (trinomial) Quadratic Binomial Quadratic Trinomial . . .more polynomials. . . 5b2 + 2 5b2 + 3b + 2 3f3 2f3 – 7 7f3 – 2f2 + f 9k4 2nd degree (quadratic) 2nd degree (quadratic) 3rd degree (cubic) 2 terms (binomial) 3 terms (trinomial) Quadratic Binomial Quadratic Trinomial . . .more polynomials. . . 5b2 + 2 5b2 + 3b + 2 3f3 2f3 – 7 7f3 – 2f2 + f 9k4 2nd degree 2 terms (quadratic) (binomial) 2nd degree 3 terms (quadratic) (trinomial) 3rd degree 1 term (cubic) (monomial) Quadratic Binomial Quadratic Trinomial . . .more polynomials. . . 5b2 + 2 5b2 + 3b + 2 3f3 2f3 – 7 7f3 – 2f2 + f 9k4 2nd degree 2 terms (quadratic) (binomial) 2nd degree 3 terms (quadratic) (trinomial) 3rd degree 1 term (cubic) (monomial) Quadratic Binomial Quadratic Trinomial Cubic Monomial . . .more polynomials. . . 5b2 + 2 5b2 + 3b + 2 3f3 2f3 – 7 7f3 – 2f2 + f 9k4 2nd degree 2 terms (quadratic) (binomial) 2nd degree 3 terms (quadratic) (trinomial) 3rd degree 1 term (cubic) (monomial) 3rd degree (cubic) Quadratic Binomial Quadratic Trinomial Cubic Monomial . . .more polynomials. . . 5b2 + 2 5b2 + 3b + 2 3f3 2f3 – 7 7f3 – 2f2 + f 9k4 2nd degree 2 terms (quadratic) (binomial) 2nd degree 3 terms (quadratic) (trinomial) 3rd degree 1 term (cubic) (monomial) 3rd degree 2 terms (cubic) (binomial) Quadratic Binomial Quadratic Trinomial Cubic Monomial . . .more polynomials. . . 5b2 + 2 5b2 + 3b + 2 3f3 2f3 – 7 7f3 – 2f2 + f 9k4 2nd degree 2 terms Quadratic (quadratic) (binomial) Binomial 2nd degree 3 terms Quadratic (quadratic) (trinomial) Trinomial 3rd degree 1 term Cubic (cubic) (monomial) Monomial 3rd degree 2 terms Cubic (cubic) (binomial) Binomial . . .more polynomials. . . 5b2 + 2 5b2 + 3b + 2 3f3 2f3 – 7 7f3 – 2f2 + f 9k4 2nd degree 2 terms Quadratic (quadratic) (binomial) Binomial 2nd degree 3 terms Quadratic (quadratic) (trinomial) Trinomial 3rd degree 1 term Cubic (cubic) (monomial) Monomial 3rd degree 2 terms Cubic (cubic) (binomial) Binomial 3rd degree (cubic) . . .more polynomials. . . 5b2 + 2 5b2 + 3b + 2 3f3 2f3 – 7 7f3 – 2f2 + f 9k4 2nd degree 2 terms Quadratic (quadratic) (binomial) Binomial 2nd degree 3 terms Quadratic (quadratic) (trinomial) Trinomial 3rd degree 1 term Cubic (cubic) (monomial) Monomial 3rd degree 2 terms Cubic (cubic) (binomial) Binomial 3rd degree 3 terms (cubic) (trinomial) . . .more polynomials. . . 5b2 + 2 5b2 + 3b + 2 3f3 2f3 – 7 7f3 – 2f2 + f 9k4 2nd degree 2 terms Quadratic (quadratic) (binomial) Binomial 2nd degree 3 terms Quadratic (quadratic) (trinomial) Trinomial 3rd degree 1 term Cubic (cubic) (monomial) Monomial 3rd degree 2 terms Cubic (cubic) (binomial) Binomial 3rd degree 3 terms Cubic (cubic) (trinomial) Trinomial . . .more polynomials. . . 5b2 + 2 5b2 + 3b + 2 3f3 2f3 – 7 7f3 – 2f2 + f 9k4 2nd degree 2 terms Quadratic (quadratic) (binomial) Binomial 2nd degree 3 terms Quadratic (quadratic) (trinomial) Trinomial 3rd degree 1 term Cubic (cubic) (monomial) Monomial 3rd degree 2 terms Cubic (cubic) (binomial) Binomial 3rd degree 3 terms Cubic (cubic) (trinomial) Trinomial 4th degree (quartic) . . .more polynomials. . . 5b2 + 2 5b2 + 3b + 2 3f3 2f3 – 7 7f3 – 2f2 + f 9k4 2nd degree 2 terms Quadratic (quadratic) (binomial) Binomial 2nd degree 3 terms Quadratic (quadratic) (trinomial) Trinomial 3rd degree 1 term Cubic (cubic) (monomial) Monomial 3rd degree 2 terms Cubic (cubic) (binomial) Binomial 3rd degree 3 terms Cubic (cubic) (trinomial) Trinomial 4th degree 1 term (quartic) (monomial) . . .more polynomials. . . 5b2 + 2 5b2 + 3b + 2 3f3 2f3 – 7 7f3 – 2f2 + f 9k4 2nd degree 2 terms Quadratic (quadratic) (binomial) Binomial 2nd degree 3 terms Quadratic (quadratic) (trinomial) Trinomial 3rd degree 1 term Cubic (cubic) (monomial) Monomial 3rd degree 2 terms Cubic (cubic) (binomial) Binomial 3rd degree 3 terms Cubic (cubic) (trinomial) Trinomial 4th degree 1 term Quartic (quartic) (monomial) Monomial . . .more polynomials. . . 9k4 – 3k2 9k4 + 3k2 + 2 v5 v5 – 7v f5 – 2f4 + f2 j7 - 2j5 + 7j3 - 8 . . .more polynomials. . . 9k4 – 3k2 9k4 + 3k2 + 2 v5 v5 – 7v f5 – 2f4 + f2 j7 - 2j5 + 7j3 - 8 4th degree (quartic) . . .more polynomials. . . 9k4 – 3k2 9k4 + 3k2 + 2 v5 v5 – 7v f5 – 2f4 + f2 j7 - 2j5 + 7j3 - 8 4th degree (quartic) 2 terms (binomial) . . .more polynomials. . . 9k4 – 3k2 9k4 + 3k2 + 2 v5 v5 – 7v f5 – 2f4 + f2 j7 - 2j5 + 7j3 - 8 4th degree (quartic) 2 terms (binomial) Quartic Binomial . . .more polynomials. . . 9k4 – 3k2 4th degree (quartic) 9k4 + 3k2 + 2 4th degree (quartic) v5 v5 – 7v f5 – 2f4 + f2 j7 - 2j5 + 7j3 - 8 2 terms (binomial) Quartic Binomial . . .more polynomials. . . 9k4 – 3k2 4th degree (quartic) 2 terms (binomial) 9k4 + 3k2 + 2 4th degree (quartic) 3 terms (trinomial) v5 v5 – 7v f5 – 2f4 + f2 j7 - 2j5 + 7j3 - 8 Quartic Binomial . . .more polynomials. . . 9k4 – 3k2 4th degree (quartic) 2 terms (binomial) Quartic Binomial 9k4 + 3k2 + 2 4th degree (quartic) 3 terms (trinomial) Quartic Trinomial v5 v5 – 7v f5 – 2f4 + f2 j7 - 2j5 + 7j3 - 8 . . .more polynomials. . . 9k4 – 3k2 4th degree (quartic) 2 terms (binomial) Quartic Binomial 9k4 + 3k2 + 2 4th degree (quartic) 3 terms (trinomial) Quartic Trinomial v5 5th degree (quintic) v5 – 7v f5 – 2f4 + f2 j7 - 2j5 + 7j3 - 8 . . .more polynomials. . . 9k4 – 3k2 4th degree (quartic) 2 terms (binomial) Quartic Binomial 9k4 + 3k2 + 2 4th degree (quartic) 3 terms (trinomial) Quartic Trinomial v5 5th degree (quintic) 1 term (monomial) v5 – 7v f5 – 2f4 + f2 j7 - 2j5 + 7j3 - 8 . . .more polynomials. . . 9k4 – 3k2 4th degree (quartic) 2 terms (binomial) Quartic Binomial 9k4 + 3k2 + 2 4th degree (quartic) 3 terms (trinomial) Quartic Trinomial v5 5th degree (quintic) 1 term Quintic (monomial) Monomial v5 – 7v f5 – 2f4 + f2 j7 - 2j5 + 7j3 - 8 . . .more polynomials. . . 9k4 – 3k2 4th degree (quartic) 2 terms (binomial) Quartic Binomial 9k4 + 3k2 + 2 4th degree (quartic) 3 terms (trinomial) Quartic Trinomial v5 5th degree (quintic) 1 term Quintic (monomial) Monomial v5 – 7v 5th degree (quintic) f5 – 2f4 + f2 j7 - 2j5 + 7j3 - 8 . . .more polynomials. . . 9k4 – 3k2 4th degree (quartic) 2 terms (binomial) Quartic Binomial 9k4 + 3k2 + 2 4th degree (quartic) 3 terms (trinomial) Quartic Trinomial v5 5th degree (quintic) 1 term Quintic (monomial) Monomial v5 – 7v 5th degree (quintic) f5 – 2f4 + f2 j7 - 2j5 + 7j3 - 8 2 terms (binomial) . . .more polynomials. . . 9k4 – 3k2 4th degree (quartic) 2 terms (binomial) Quartic Binomial 9k4 + 3k2 + 2 4th degree (quartic) 3 terms (trinomial) Quartic Trinomial v5 5th degree (quintic) 1 term Quintic (monomial) Monomial v5 – 7v 5th degree (quintic) f5 – 2f4 + f2 j7 - 2j5 + 7j3 - 8 2 terms (binomial) Quintic Binomial . . .more polynomials. . . 9k4 – 3k2 4th degree (quartic) 2 terms (binomial) Quartic Binomial 9k4 + 3k2 + 2 4th degree (quartic) 3 terms (trinomial) Quartic Trinomial v5 5th degree (quintic) 1 term Quintic (monomial) Monomial v5 – 7v 5th degree (quintic) f5 – 2f4 + f2 5th degree (quintic) j7 - 2j5 + 7j3 - 8 2 terms (binomial) Quintic Binomial . . .more polynomials. . . 9k4 – 3k2 4th degree (quartic) 2 terms (binomial) Quartic Binomial 9k4 + 3k2 + 2 4th degree (quartic) 3 terms (trinomial) Quartic Trinomial v5 5th degree (quintic) 1 term Quintic (monomial) Monomial v5 – 7v 5th degree (quintic) 2 terms (binomial) f5 – 2f4 + f2 5th degree (quintic) 3 terms (trinomial) j7 - 2j5 + 7j3 - 8 Quintic Binomial . . .more polynomials. . . 9k4 – 3k2 4th degree (quartic) 2 terms (binomial) Quartic Binomial 9k4 + 3k2 + 2 4th degree (quartic) 3 terms (trinomial) Quartic Trinomial v5 5th degree (quintic) 1 term Quintic (monomial) Monomial v5 – 7v 5th degree (quintic) 2 terms (binomial) Quintic Binomial f5 – 2f4 + f2 5th degree (quintic) 3 terms (trinomial) Quintic Trinomial j7 - 2j5 + 7j3 - 8 . . .more polynomials. . . 9k4 – 3k2 4th degree (quartic) 2 terms (binomial) Quartic Binomial 9k4 + 3k2 + 2 4th degree (quartic) 3 terms (trinomial) Quartic Trinomial v5 5th degree (quintic) 1 term Quintic (monomial) Monomial v5 – 7v 5th degree (quintic) 2 terms (binomial) Quintic Binomial f5 – 2f4 + f2 5th degree (quintic) 3 terms (trinomial) Quintic Trinomial j7 - 2j5 + 7j3 - 8 7th degree (7th degree) . . .more polynomials. . . 9k4 – 3k2 4th degree (quartic) 2 terms (binomial) Quartic Binomial 9k4 + 3k2 + 2 4th degree (quartic) 3 terms (trinomial) Quartic Trinomial v5 5th degree (quintic) 1 term Quintic (monomial) Monomial v5 – 7v 5th degree (quintic) 2 terms (binomial) Quintic Binomial f5 – 2f4 + f2 5th degree (quintic) 3 terms (trinomial) Quintic Trinomial j7 - 2j5 + 7j3 - 8 7th degree (7th degree) 4 terms (polynomial with 4 terms) . . .more polynomials. . . 9k4 – 3k2 4th degree (quartic) 2 terms (binomial) Quartic Binomial 9k4 + 3k2 + 2 4th degree (quartic) 3 terms (trinomial) Quartic Trinomial v5 5th degree (quintic) 1 term Quintic (monomial) Monomial v5 – 7v 5th degree (quintic) 2 terms (binomial) Quintic Binomial f5 – 2f4 + f2 5th degree (quintic) 3 terms (trinomial) Quintic Trinomial j7 - 2j5 + 7j3 - 8 7th degree (7th degree) 4 terms (polynomial with 4 terms) 7th degree polynomial with 4 terms Can you name them now? Give it a try. . . copy and classify (2CLASSIFICATION columns): POLYNOMIAL / NAME 1. 5x2 – 2x + 3 2. 2z + 5 3. 7a3 + 4a – 12 4. -15 5. 27x8 + 3x5 – 7x + 4 6. 9x4 – 3 7. 10x – 185 8. 18x5 1. Quadratic Trinomial 2. Linear Binomial 3. Cubic Trinomial 4. Constant Monomial 5. 8th Degree Polynomial with 4 terms. 6. Quartic Binomial 7. Linear Binomial 8. Quintic Monomial . . .more practice. . . 9. 19z25 9. 25th degree Monomial 10. 6n7 + 3n5 – 4n 10. 7th degree Trinomial 11. 189 11. Constant Monomial 12. 15g5 + 3g4 – 3g – 7 12. Quintic polynomial with 4 terms 9 13. 3x + 12 th degree Binomial 13. 9 14. 6z3 – 2z2 + 4z + 7 14. Cubic polynomial 15. 9c with 4 terms. 16. 15b2 – b + 51 15. Linear Monomial 16.Quadratic Trinomial Create-A-Polynomial Activity: You will need one sheet of paper, a pencil, and your notes on Classifying Polynomials. You will be given the names of some polynomials. YOU must create a correct matching example, but REMEMBER: Answers must be in standard form. Each problem can only use one variable throughout the problem (see examples in notes from powerpoint). All variable powers in each example must have DIFFERENT exponents (again, see examples in notes from powerpoint). There can only be one constant in each problem. You will need two columns on your paper. Left column title: Polynomial Name Right column title: Polynomial in Standard Form. Create - A – Polynomial Activity: Copy each polynomial name. Create your own polynomial to match ( 2 columns). Partner/Independent work (pairs). 1. 6th degree polynomial with 4 terms. 2. Linear Monomial 3. Quadratic Trinomial 4. Constant Monomial 5. Cubic Binomial 6. Quartic Monomial 7. Quintic polynomial with 5 terms. 8. Linear Binomial 9. Cubic Trinomial 10. Quadratic Binomial 11. Quintic Trinomial 12. Constant Monomial 13. Cubic Monomial 14. 10th degree polynomial with 6 terms 15. 8th degree Binomial 16. 20th degree Trinomial Warm-Up 9.1 Adding and Subtracting Polynomials Adding Polynomials in horizontal form: (x2 + 2x + 5) + (3x2 + x + 12) = 4x2+ 3x + 17 in vertical form: x2 + 2x + 5 + 3x2 + x + 12 = 4x2+ 3x + 17 9.1 Adding and Subtracting Polynomials x2 +x+1 + 2x2 + 3x + 2 3x2 + 4x + 3 3x2 – 4x + 8 + 2x2 – 7x – 5 5x2 – 11x + 3 Simplify: (4b2 + 2b + 1) + (7b2 + b – 3) 11b2 + 3b – 2 a2 + 8a – 5 + 3a2 + 2a – 7 4a2 + 10a – 12 9.1 Adding and Subtracting Polynomials 7d2 + 7d + 2d2 + 3d z2 + 5z + 4 + 2z2 - 5 9d2 + 10d 3z2 + 5z – 1 Simplify: (2x2 – 3x + 5) + (4x2 + 7x – 2) 6x2 + 4x + 3 (a2 + 6a – 4) + (8a2 – 8a) 9a2 – 2a – 4 9.1 Adding and Subtracting Polynomials 3d2 + 5d – 1 + (–4d2 – 5d + 2) –d2+ 1 + 7p2 + 5 (–5p2 – 2p + 3) 2p2 – 2p + 8 Simplify: (3x2 + 5x) + (4 – 6x – 2x2) x2 – x + 4 (x3 + x2 + 7) + (2x2 + 3x – 8) x3 + 3x2 + 3x – 1 9.1 Adding and Subtracting Polynomials 2x3 x2 + –4 + 3x2 – 9x + 7 2x3 + 4x2 – 9x + 3 4p2 + 5p + (-2p + p + 7) 4p2 + 4p + 7 Simplify: 5y2 – 3y + 8 + 4y3 – 9 4y3 + 5y2 – 3y – 1 (8cd – 3d + 4c) + (-6 + 2cd – 4d) 4c – 7d + 10cd – 6 9.1 Adding and Subtracting Polynomials Simplify: (12y3 + y2 – 8y + 3) + (6y3 – 13y + 5) = 18y3 + y2 – 21y + 8 (7y3 + 2y2 – 5y + 9) + (y3 – y2 + y – 6) = 8y3 + y2 – 4y + 3 (6x5 + 3x3 – 7x – 8) + (4x4 – 2x2 + 9) = 6x5+ 4x4 + 3x3 – 2x2 – 7x + 1 9.1 Adding and Subtracting Polynomials Subtracting Polynomials (Note: subtraction is adding the opposite) (5x2 + 10x + 2) + (–x2) + (3x) + (–12) = 4x2+ 13x – 10 (5x2 + 10x + 2) – (x2 – 3x + 12) 5x2 + 10x + 2 –(x2 – 3x + 12) = 5x2 + 10x + 2 = –x2 + 3x – 12 4x2 + 13x – 10 9.1 Adding and Subtracting Polynomials Simplify: (3x2 – 2x + 8) – (x2 – 4) 3x2 – 2x + 8 + (–x2) + 4 = 2x2 – 2x + 12 (10z2 + 6z + 5) – (z2 – 8z + 7) 10z2 + 6z + 5 + (–z2) + 8z + (–7) = 9z2 + 14z – 2 9.1 Adding and Subtracting Polynomials Simplify: 7a2 – 2a – (5a2 + 3a) 2a2 – 5a 4x2 + 3x + 2 – (2x2 – 3x – 7) 2x2 + 6x + 9 – 3x2 – 7x + 5 (x2 + 4x + 7) 2x2 – 11x – 2 7x2 – x + 3 – (3x2 – x – 7) 4x2 + 10 9.1 Adding and Subtracting Polynomials Simplify: 3x2 – 2x + 10 – (2x2 + 4x – 6) x2 – 6x + 16 3x2 – 5x + 3 – (2x2 – x – 4) x2 – 4x + 7 2x2 + 5x – (x2 – 3) x2 + 5x + 3 4x2 – x + 6 – (3x2 – 4) x2 – x + 10 9.1 Adding and Subtracting Polynomials Simplify: (4x5 + 3x3 – 3x – 5) – (– 2x3 + 3x2 + x + 5) = 4x5 + 5x3 – 3x2 – 4x – 10 (4d4 – 2d2 + 2d + 8) – (5d3 + 7d2 – 3d – 9) = 4d4 – 5d3 – 9d2 + 5d + 17 (a2 + ab – 3b2) – (b2 + 4a2 – ab) = –3a2 + 2ab – 4b2 9.1 Adding and Subtracting Polynomials Simplify by finding the perimeter: 6c + 3 A c2 + 1 4c2 + 2c + 5 A= 5c2 x2 + x + 8c + 9 2x2 B 2x2 x2 + x B = 6x2 + 2x 9.1 Adding and Subtracting Polynomials Simplify by finding the perimeter: 3x2 – 5 x x2 C +7 2d2 + d – 4 3d2 – 5d D 3d2 – 5d d2 + 7 C= 14 8x2 – 10x + D = 9d2 – 9d + 3 9.1 Adding and Subtracting Polynomials Simplify by finding the perimeter: a+1 E 3x2 – 5x a3 + 2a 2a3 + a + 3 E = 3a3 + 4a + 4 x2 + 3 F F = 8x2 – 10x + 6 Multiplying a Monomial by a Polynomial Definitions: A Monomial is a single term (mono means ‘one’) Examples of monomials: 2, -5, 3x, 5x2y, xyz Notice that a monomial can be made up of numbers, variables, or both. A Polynomial is made up of two or more terms (Terms are separated by plus or minus signs) Examples of polynomials: 2x + 2, x2 – 5y, 3x + 2y - 9 TWO TERMS THREE TERMS TWO TERMS Some polynomials have special names: Binomials have exactly two terms (Bi means two) Trinomials have exactly three terms (Tri means three) Multiplying a Monomial by a Polynomial We can multiply a monomial and a polynomial by using the distributive property. Example: 2x( 2 x – 3x + 5) Whenever you have anything sitting in front of parentheses like this 2x is doing, you are supposed to multiply the term in front by each term on the inside. (This is called the “distributive property” because you are distributing the outside term to each of the inside terms) Now we’ll solve it! 2x • x 2 = 2x 3 2x • -3x = -6x 2 2x • 5 = 10x Answer: 2x 3 – 6x 2 + 10x Polynomial Multiplication using boxes (windows You can multiply any two polynomials using this method. FIRST DRAW A BOX THE RIGHT SIZE 1) Count the number of terms in each polynomial examples: (x + 3) has two terms (x2 – 2x + 7) has three terms 2) Draw a box with the number of rows and columns matching the number of terms in the two polynomials. A. (x – 6)(x + 1) 2 rows 2 columns B. (x2 – 2x + 7) (x – 1) 3 rows 2 columns 3) Once you have drawn the box, you can start multiplying Two-dimensional Polynomial Multiplication 1) Place one binomial on top and the other on the left side (x - 4) (x + 3) 2) Now multiply the outside terms in each row and each column, placing the answers in the box. x x +3 x2 +3x -4 -4x -12 3) Combine the like terms = -1x (different signs: subtract) ANSWER: x2 - x - 12 4) Write the answer starting with the term in the top left-hand corner (inside the box), the diagonal terms are next and the constant (bottom right number) is last. Two-dimensional Polynomial Multiplication 1) Place one binomial on top and the other on the left side (x + 6) (x + 7) 2) Now multiply the outside terms in each row and each column, placing the answers in the box. x x +7 x2 +7x +6 +6x +42 3) Combine like terms = +13x (same signs: add) ANSWER: x2 + 13x + 42 4) Write the answer starting with the term in the top left-hand corner (inside the box), the diagonal terms are next and the constant (bottom right number) is last. Two-dimensional Polynomial Multiplication 1) Place one binomial on top and the other on the left side (x - 4y) (x - 2y) 2) Now multiply the outside terms in each row and each column, placing the answers in the box. x -2y x x2 -4y -4xy 3) Combine like terms = -6xy (same signs: add) -2xy +8y2 ANSWER: x2 - 6xy + 8y2 4) Write the answer starting with the term in the top left-hand corner (inside the box), the diagonal terms are next and the constant (bottom right number) is last. Other Combinations x2 +2x x +2 (x2 + 2x - 3)(x + 2) x3 +2x2 2 +2x +4x -3 -3x -6 Answer: x 3 + 4x 2 + x - 6 Like Terms = +4x2 (same signs: add) Like Terms = +x (different signs: subtract) 1) Multiply each row by each column 2) Combine like terms on the diagonals 3) Write the answer starting from the top left and moving to the bottom right. (x + 2)(x2 + 5x - 3) Other Combinations 2 x +5x -3 Like Terms = +7x2 x +2 The like terms always land on this same diagonal x3 2 +5x 2 +2x +10x Answer: -3x -6 x 3 + 7x 2 + 7x - 6 Like Terms = +7x 1) Multiply each row by each column 2) Combine like terms on the diagonals 3) Write the answer starting from the top left and moving to the bottom right. Multiplying 2 Trinomials(x2 - 3x + 6)(x2 + 4x - 2) 2 x +6 -3x 2 x x4 -3x3 +6x2 +4x +4x3 -12x2 +24x -2 -2x2 +6x ANSWER: x4 1) Multiply each row by each column 2) Combine like terms on the diagonals 3) 3) Write the answer starting from the top left and moving to the bottom right. -12 + x3 - 8x2 + 30x -12 Practice Problems: (Hit enter to see the answers) 1) (x + 7)(x – 3) 5) (4x – 3)(x2 + 2x – 1) 2) (x – 2)(x – 5) 6) (2x2 – 6x + 1)(3x2 – 4x – 2) 3) (x + 2y)(3x + y) 7) (2x2 + 3x – 4)(x + 7) 4) (x + 3)(x – 3) 8) (2 + x)(3 + x) Answers: 1) x2 + 4x – 21 2) x2 – 7x + 10 3) 3x2 + 7xy + 2y2 4) x2 - 9 5) 4x3 + 5x2 – 10x + 3 6) 6x4 – 26x3 + 23x2 + 8x - 2 7) 2x3 +17x2 + 17x – 28 8) 6 + 5x + x2 Warm-Up There are three techniques you can use for multiplying polynomials. The best part about it is that they are all the same! Huh? Whaddaya mean? It’s all about how you write it…Here they are! 1)Distributive Property 2)FOIL 3)Box Method Sit back, relax (but make sure to write this down), and I’ll show ya! 1) Multiply. (2x + 3)(5x + 8) Using the distributive property, multiply 2x(5x + 8) + 3(5x + 8). 10x2 + 16x + 15x + 24 Combine like terms. 10x2 + 31x + 24 A shortcut of the distributive property is called the FOIL method. The FOIL method is ONLY used when you multiply 2 binomials. It is an acronym and tells you which terms to multiply. 2) Use the FOIL method to multiply the following binomials: (y + 3)(y + 7). (y + 3)(y + 7). F tells you to multiply the FIRST terms of each binomial. y2 (y + 3)(y + 7). O tells you to multiply the OUTER terms of each binomial. y2 + 7y (y + 3)(y + 7). I tells you to multiply the INNER terms of each binomial. y2 + 7y + 3y (y + 3)(y + 7). L tells you to multiply the LAST terms of each binomial. y2 + 7y + 3y + 21 Combine like terms. y2 + 10y + 21 Remember, FOIL reminds you to multiply the: First terms Outer terms Inner terms Last terms The third method is the Box Method. This method works for every problem! 3x Here’s how you do it. Multiply (3x – 5)(5x + 2) Draw a box. Write a polynomial on the top and side of a box. It does not matter which goes where. This will be modeled in the next problem along with FOIL. 5x +2 -5 3) Multiply (3x - 5)(5x + 2) First terms: 15x2 Outer terms: +6x Inner terms: Last terms: -25x Combine -10 like terms. 15x2 - 19x – 10 3x 5x -5 15x2 -25x +2 +6x -10 You have 3 techniques. Pick the one you like the best! 4) Multiply (7p - 2)(3p - 4) First terms: 21p2 Outer terms: -28p Inner terms: -6p Last terms: +8 Combine like terms. 21p2 – 34p + 8 7p 3p -2 21p2 -6p -4 -28p +8 Multiply (y + 4)(y – 3) 1. y2 + y – 12 2. y2 – y – 12 3. y2 + 7y – 12 4. y2 – 7y – 12 5. y2 + y + 12 6. y2 – y + 12 7. y2 + 7y + 12 8. y2 – 7y + 12 Multiply (2a – 3b)(2a + 4b) 1. 4a2 + 14ab – 12b2 2. 4a2 – 14ab – 12b2 3. 4a2 + 8ab – 6ba – 12b2 4. 4a2 + 2ab – 12b2 5. 4a2 – 2ab – 12b2 5) Multiply (2x - 5)(x2 - 5x + 4) You cannot use FOIL because they are not BOTH binomials. You must use the distributive property. 2x(x2 - 5x + 4) - 5(x2 - 5x + 4) 2x3 - 10x2 + 8x - 5x2 + 25x - 20 Group and combine like terms. 2x3 - 10x2 - 5x2 + 8x + 25x - 20 2x3 - 15x2 + 33x - 20 5) Multiply (2x - 5)(x2 - 5x + 4) You cannot use FOIL because they are not BOTH binomials. You must use the distributive property or box method. x2 -5x +4 2x 2x3 -10x2 +8x -5 -5x2 +25x -20 Almost done! Go to the next slide! 5) Multiply (2x - 5)(x2 - 5x + 4) Combine like terms! x2 -5x +4 2x 2x3 -10x2 +8x -5 -5x2 +25x -20 2x3 – 15x2 + 33x - 20 Multiply (2p + 1)(p2 – 3p + 4) 1. 2p3 + 2p3 + p + 4 2. y2 – y – 12 3. y2 + 7y – 12 4. y2 – 7y – 12 Rules for Factoring Polynomials A “Difference of Squares” is a binomial (*2 terms only*) and it factors like this: 2 2 a − b = (a + b)(a − b) Factoring a polynomial means expressing it as a product of other polynomials. Factoring Method #1 Factoring polynomials with a common monomial factor (using GCF). **Always look for a GCF before using any other factoring method. Steps : 1. Find the greatest common factor (GCF). 2. Divide the polynomial by the GCF. The quotient is the other factor. 3. Express the polynomial as the product of the quotient and the GCF. Example : 3 2 2 6c d − 12c d + 3cd GCF = 3cd Step 1: Step 2: Divide by GCF 2 3 2 (6c d − 12c d + 3cd) ÷ 3cd = 2 2c − 4cd + 1 The answer should look like this: 3 2 2 Ex: 6c d − 12c d + 3cd 2 = 3cd(2c − 4cd + 1) Factor these on your own looking for a GCF. 3 2 1. 6x + 3x − 2 = 3 x 2 x ( + x − 4) 12x 2 2. 5x − 10x + 35 = 5( x2 − 2 x + 7) 3. 16x y z − 8x y z + 12xy z 3 4 2 2 3 = 4 xy z ( 4 x y − 2 xz + 3 yz ) 2 2 2 2 3 2 Factoring Method #2 Factoring polynomials that are a difference of squares. To factor, express each term as a square of a monomial then apply the rule... 2 2 a − b = (a + b)(a − b) 2 Ex: x − 16 = 2 2 x − 4 = (x + 4)(x − 4) Here is another example: 1 2 x − 81 = 49 2 1  2  1 x + 9  1 x − 9 x − 9 =  7  7 7  Try these on your own: ( x + 11)( x − 11) 1. x − 121 = 2 2. 9y − 169x 2 2 =− ( 3 y 13 x )( 3 y + 13 x ) 3. x − 16 = ( x − 2 )( x + 2 ) ( x + 4 ) 4 Be careful! 2 Sum and Difference of Cubes: a + b = (a + b)(a − ab + b 3 3 3 3 a −b ) = (a − b)(a + ab + b ) 2 2 2 2 Write each monomial as a cube and apply either of the rules. Example : Rewrite as 3 cubes3 3 x + 64 = (x + 4 ) Apply the rule for sum of cubes: a + b = (a + b)(a − ab + b 3 2 3 2 2 = (x + 4)(x − x ⋅4 + 4 ) 2 = (x + 4)(x − 4x + 16) 2 ) 3 Rewrite as cubes 3 Ex: 8y − 125 3 = ((2y) − 5 ) Apply the rule for difference of cubes: a − b = (a − b)(a + ab + b 3 3 2 ( 2 ) = (2y − 5) (2y) + 2y ⋅ 5 + (5) 2 2 = (2y − 5)(4y + 10y + 25) 2 ) Factoring Method #3 Factoring a trinomial in the form: 2 ax + bx + c Factoring a trinomial: 2 ax + bx + c 1. Write two sets of parenthesis, ( )( ). These will be the factors of the trinomial. 2. Product of first terms of both binomials 2 equal first term of the must (ax ) trinomial. Next Factoring a trinomial: 2 ax + bx + c 3. The product of last terms of both binomials must equal last term of the trinomial (c). 4. Think of the FOIL method of multiplying binomials, the sum of the outer and the inner products must equal the middle term (bx). 2 x − 6x + 8 Example : (x (x )( x -2 )( x Factors of +8: -4 ) ) x⋅ x = x O + I = bx ? 1&8 1x + 8x = 9x 2&4 2x + 4x = 6x -1 & -8 -1x - 8x = -9x -2 & -4 -2x - 4x = -6x 2 2 x − 6x + 8 = (x − 2)(x − 4) Check your answer by using FOIL F O I 2 L (x − 2)(x − 4) = x − 4x − 2x + 8 2 = x − 6x + 8 Lets do another example: 2 6x − 12x − 18 Don’t Forget Method #1. Always check for GCF before you do anything else. 2 6(x − 2x − 3) 6(x − 3)(x + 1) Find a GCF Factor trinomial When a>1 and c<1, there may be more combinations to try! Example : Step 1: 2 6 x + 13 x − 5 2 Find the factors of 6x : 3x ⋅ 2x 6x ⋅ x 2 Example : 6 x + 13 x − 5 Step 2: 5 -1 -5  1  -1  5    1  -5  Find the factors of -5: Order can make a difference! 2 Example : 6 x + 13 x − 5 Step 3: Place the factors inside the parenthesis until O + I = bx. Try: F 2 O (6x − 1)(x + 5) I L 6x + 30x − x − 5 O + I = 30 x - x = 29x This doesn’t work!! 2 Example : 6 x + 13 x − 5 Switch the order of the second terms and try again. (6x + 5)(x − 1) F 2 O I L 6x − 6x + 5x − 5 O + I = -6x + 5x = -x This doesn’t work!! Try another combination: Switch to 3x and 2x (3x − 1)(2x + 5) F 2 O I L 6x + 15x − 2x − 5 O+I = 15x - 2x = 13x IT WORKS!! 6x + 13x − 5 = (3x − 1)(2x + 5) 2 Factoring Technique #3 Factoring a perfect square trinomial in the form: continued a + 2ab + b = (a + b) 2 a − 2ab + b = (a − b) 2 2 2 2 2 Perfect Square Trinomials can be factored just like other trinomials (guess and check), but if you recognize the perfect squares pattern, follow the formula! a + 2ab + b = (a + b) 2 a − 2ab + b = (a − b) 2 2 2 2 2 2 Ex: x + 8x + 16 2 2 (4 ) (x ) a Does the middle term fit the pattern, 2ab? b 2 ⋅ x ⋅ 4 = 8x Yes, the factors are (a + b)2 : 2 x + 8x + 16 = (x + 4) 2 2 Ex: 4x − 12x + 9 2 2 (3) (2x ) a Does the middle term fit the pattern, 2ab? b 2 ⋅2x ⋅ 3 = 12x Yes, the factors are (a - b)2 : 4x − 12x + 9 = (2x − 3) 2 2 Factoring Technique #4 Factoring By Grouping for polynomials with 4 or more terms Factoring By Grouping 1. Group the first set of terms and last set of terms with parentheses. 2. Factor out the GCF from each group so that both sets of parentheses contain the same factors. 3. Factor out the GCF again (the 3 b − 3b Step 1: Group Example 1: = (b − 3b 3 2 2 + 4 b − 12 )+ (4b − 12 ) Step 2: Factor out GCF from each 2 group = b (b − 3) + 4(b − 3) Step 3: Factor out GCF 2 again = (b − 3)(b + 4) 3 Example 2: 2 2 x − 16 x − 8 x + 64 = 2(x − 8x − 4x + 32 ) 2 3 = 2 (x − 8x )+ (−4x + 32) 3 ( 2 = 2(x (x − 8) + −4(x − 8)) 2 = 2 (x − 8)(x − 4 ) = 2((x − 8)(x − 2)(x + 2)) ( 2 ) ) Try these on your own: 2 1. x − 5x − 6 2 2. 3x + 11x − 20 3 3. x + 216 4. 8x − 8 3 5. 3x − 6x − 24x 3 2 Answers: 1. (x − 6)(x + 1) 2. (3x − 4)(x + 5) 2 3. (x + 6)(x − 6x + 36) 4. 8(x − 1)(x + x + 1) 2 5. 3x(x − 4)(x + 2)