Questions and Answers #4 Polynomial Arithmetic

Questions and Answers Sheet 4 Polynomial Arithmetic Question #1 Indicate whether the expression defines a polynomial function. 𝑃(𝑥) = −𝑥 2 + 3𝑥 + 3 polynomial or not a polynomial If it is a polynomial function, identify the following. (If it is not a polynomial function, enter DNE for all three answers.) (a) Identify the leading coefficient. (b) Identify the constant term. (c) State the degree. Answer: Step 1 Definition used: ''A polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication and non-negative integer exponents of variables. The polynomial equation is of the form 𝑐𝑛 𝑥 𝑛 + 𝑐𝑐−1 𝑥 𝑛−1 + ⋯ + 𝑐2 𝑥 2 + 𝑐1 𝑥 + 𝑐0 Here, 𝑐𝑛 is the leading coefficient, n is the highest degree and the constant is 𝑐0 .'' Step 2 a) The given function is 𝑃(𝑥) = −𝑥 2 + 3𝑥 + 3. By the definition of polynomial, the above function is quadratic polynomial. The leading coefficient is -1. Step 3 b) By the definition of polynomial equation, the constant term is 3. Step 4 c) By the definition of polynomial equation, the degree is 2. Question #2 Exponentials of an Arithmetic Sequence, If 𝑎1 , 𝑎2 , 𝑎3 , ⋯ is an arithmetic sequence with common difference d, show that the sequence 10𝑎1 , 10𝑎2 , 10𝑎3 , ⋯ is a geometric sequence, and find the common ratio. Answer: Given, 𝑎1 , 𝑎2 , 𝑎3 , ⋯ is a n arithmetic sequence with common difference d. Now, 𝑎2 = 𝑎1 + 𝑑, 𝑎3 = 𝑎1 + 2𝑑, − − − − −[∴ 𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑]. Now consider the sequence, 10𝑎1 , 10𝑎2 , 10𝑎3 , ⋯ Here, 10𝑎2 10𝑎1 +𝑑 = = 10𝑑 10𝑎3 10𝑎2 = +2𝑑 = 102𝑑 10𝑑 = 10𝑑 10𝑎1 +𝑑 10𝑎1 10𝑎2 10𝑎1 Clearly, second term first term = third term = 10𝑑 second term 𝑎1 𝑎2 Therefore 10 , 10 , 10𝑎3 , ⋯ is a geometric sequence. The common ratio is 10𝑑 . Question #3 To find: The square of a polynomial (4𝑎 − 3𝑏)2 using indicated operation. Answer: Concept: Multiplication of polynomials: 1) To find the product of two polynomials, distribute each term of one polynomial, multiplying by each term of other polynomial. 2) Another method is to write such a product vertically, similar to the method used in arithmetic for multiplying whole numbers. 3) The FOIL method is a convenient way to find the product of two binomials. The memory aid FOIL (for First, Outer, Inner, Last) gives the pairs of terms to be multiplied when distributing each term of the first binomial, multiplying by each term of the second binomial. 4) Using the Special Products like Square of a Binomial: (𝑎 + 𝑏)2 = 𝑎2 + 2𝑎𝑏 + 𝑏 2 and (𝑎 − 𝑏)2 = 𝑎2 − 2𝑎𝑏 + 𝑏 2 Product of the Sum and Difference of Two Terms: 𝑎2 − 𝑏 2 = (𝑎 + 𝑏)(𝑎 − 𝑏) Calculation: Given that (4𝑎 − 3𝑏)2 To find the product of two, apply Square of a Binomial (𝑎 − 𝑏)2 = 𝑎2 − 2𝑎𝑏 + 𝑏 2 = (4𝑎)2 − 2(4𝑎)(3𝑏) + (3𝑏)2 = 16𝑎2 − 24𝑎𝑏 + 9𝑏 2 [Expand exponentials by power rule 2 and multiply] Final statement: Therefore, (4𝑎 − 3𝑏)2 = 16𝑎2 − 24𝑎𝑏 + 9𝑏 2 Question #4 To find: The square of a polynomial (3𝑘 − 5𝑚)2 using indicated operation Answer: Concept: Multiplication of polynomials: 1) To find the product of two polynomials, distribute each term of one polynomial, multiplying by each term of other polynomial. 2) Another method is to write such a product vertically, similar to the method used in arithmetic for multiplying whole numbers. 3) The FOIL method is a convenient way to find the product of two binomials. The memory aid FOIL (for First, Outer, Inner, Last) gives the pairs of terms to be multiplied when distributing each term of the first binomial, multiplying by each term of the second binomial. 4) Using the Special Products like Square of a Binomial: (𝑎 − 𝑏)2 = 𝑎2 − 2𝑎𝑏 + 𝑏 2 and (𝑎 − 𝑏)2 = 𝑎2 − 2𝑎𝑏 + 𝑏 2 Product of the Sum and Difference of Two Terms: 𝑎2 − 𝑏 2 = (𝑎 + 𝑏)(𝑎 − 𝑏) Calculation: Given that (3𝑘 − 5𝑚)2 To find the product of two, apply Square of a Binomial (𝑎 − 𝑏)2 = 𝑎2 − 2𝑎𝑏 + 𝑏 2 = (3𝑘)2 − 2(3𝑘)(5𝑚) + (5𝑚)2 = 9𝑘 2 − 30𝑘𝑚 + 25𝑚2 [Expand exponentials by power rule 2 and multiply] Final statement: Therefore, (3𝑘 − 5𝑚)2 = 9𝑘 2 − 30𝑘𝑚 + 25𝑚2 Question #5 To calculate: The coefficient and the degree of each term of the polynomial 8 − 𝑥 2 𝑦 4 + 𝑦 7 and then find the degree of the polynomial. Answer: Consider the polynomial. 8 − 𝑥 2𝑦4 + 𝑦7 Since, the terms are separated by addition signs, the terms are listed as follows: 8, − 𝑥 2 𝑦 4 and 𝑦 7 Thus, the terms of the polynomial 8 − 𝑥 2 𝑦 4 + 𝑦 7 are 8, −𝑥 2 𝑦 4 and 𝑦 7 . The coefficients of each term of the polynomial: 8 − 𝑥 2 𝑦 4 + 𝑦 7 are 8, −𝑥 2 and 1. The degree of each term of the polynomial 8 − 𝑥 2 𝑦 4 + 𝑦 7 are 0.4, and 7. The degree of the polynomial 8 − 𝑥 2 𝑦 4 + 𝑦 7 is 7. Question #6 To calculate: The coefficient and the degree of each term of the polynomial −𝑥 2 𝑦 + 4𝑦 3 − 2𝑥𝑦 and then find the degree of the polynomial. Answer: Take factor of y on first term and third term: −𝑥 2 𝑦 + 4𝑦 3 − 2𝑥𝑦 = 4𝑦 3 − (𝑥 2 + 2𝑥)𝑦 Since, the terms are saparated by addition signs, the terms are listed as follows: 4𝑦 3 and −(𝑥 2 + 2𝑥)𝑦 Thus, the terms of the polynomial −𝑥 2 𝑦 + 4𝑦 3 − 2𝑥𝑦 are 4𝑦 3 and −(𝑥 2 + 2𝑥)𝑦 The coefficients of each term of the polynomial −𝑥 2 𝑦 + 4𝑦 3 − 2𝑥𝑦 is 4 and −(𝑥 2 + 2𝑥) The degree of each term of the polynomial −𝑥 2 𝑦 + 4𝑦 3 − 2𝑥𝑦 is 3 and 1. The degree of the polynomial −𝑥 2 𝑦 + 4𝑦 3 − 2𝑥𝑦 is 3. Question #7 To find: The product of two polynomials (2r+11s) and (4r-9s) using indicated operation Answer: Step 1 Given that (2𝑟 + 11𝑠)(4𝑟 − 9𝑠) To find the product of teo polynomials (2r+11s) and (4r-9s) , is to distribute each term of (2r+11s) , multiplying by each term of (4r-9s) = 2𝑟(4𝑟 − 9𝑠) + 11𝑠(4𝑟 − 9𝑠) = 8𝑟 2 − 18𝑟𝑠 + 44𝑠𝑟 − 99𝑠 2 [Distributive law] = 8𝑟 2 + (44 − 18)𝑠𝑟 − 99𝑠 2 [Grouping like terms] = 8𝑟 2 + 26𝑠𝑟 − 99𝑠 2 [adding coefficients of like terms] Step 2 Foil method: (2𝑟 + 11𝑠)(4𝑟 − 9𝑠) = (2𝑟)(4𝑟) + (2𝑟)(−9𝑠) + (11𝑠)(4𝑟) + (11𝑠)(−9𝑠) = 8𝑟 2 − 18𝑟𝑠 + 44𝑠𝑟 − 99𝑠 2 [Multiply] = 8𝑟 2 + (44 − 18)𝑠𝑟 − 99𝑠 2 [Grouping like terms] = 8𝑟 2 + 26𝑠𝑟 − 99𝑠 2 [adding coefficients of like terms] Therefore: (2𝑟 + 11𝑠)(4𝑟 − 9𝑠) = 8𝑟 2 + 26𝑠𝑟 − 99𝑠 2 Question #8 To calculate: The coefficient and the degree of each term of the polynomial 18𝑥 2 + 36𝑥 9 − 7𝑥 + 3 and then find the degree of the polynomial. Answer: Consider the polynomial. 18𝑥 3 + 36𝑥 9 − 7𝑥 + 3 Since, the terms are saparated by addition signs, the terms are listed as follows: 18𝑥 3 , 36𝑥 9 , −7𝑥 and 3 Thus, the terms of the polynomial 18𝑥 3 + 36𝑥 9 − 7𝑥 + 3 are 18𝑥 3 , 36𝑥 9 , −7𝑥 and 3. The coefficients of each term of the polynomial 18𝑥 3 + 36𝑥 9 − 7𝑥 + 3 are 18, 36, -7 and 3. The degree of each term of the polynomial 18𝑥 3 + 36𝑥 9 − 7𝑥 + 3 abd 3, 9, 1, and 0. The degree of the polynomial 18𝑥 3 + 36𝑥 9 − 7𝑥 + 3 is 9. Question #9 Determine whether the following function is a polynomial function. If the function is a polynomial function, state its degree. If it is not, tell why not. Write the polynomial in standard form. Then identify the leading term and the constant term. 4 𝑓(𝑥) = 7 − 2 𝑥 Answer: Step 1 Given: 𝑓(𝑥) = 7 − 4 𝑥2 𝑓(𝑥) = 7 − 4𝑥 −2 A polynomial will be of the form 𝑎0 + 𝑎1 𝑥 + 𝑎2 𝑥 2 + 𝑎3 𝑥 3 + ⋯ 𝑎𝑛 𝑥 𝑛 where 𝑛 ∈ 𝑁 Step 2 7 − 4𝑥 −2 is not a polynomial as the exponent of x is -2, which is not a natural number. Since, 𝑓(𝑥) = 7 − 4𝑥 −2 is not a pklinomial, its degree is not defined. Question #10 To calculate: The product of [𝑥 − (1 + √2)][𝑥 − (1 − √2)]. Answer: Step 1 Given: Algebraic expression in multiplication form, [𝑥 − (1 + √2)][𝑥 − (1 − √2)] Formula Used: Polynomial identity, (𝑎 − 𝑏)(𝑎 + 𝑏) = 𝑎2 − 𝑏 2 (𝑎 − 𝑏)2 = 𝑎2 2𝑎𝑏 + 𝑏 2 Associative property, (a+b)+c=a+(b+c) Step 2 If P(x) represent the given expression, then 𝑃(𝑥) = [𝑥 − (1 + √2)][𝑥 − (1 − √2)] 𝑃(𝑥) = (𝑥 − 1 − √2)(𝑥 − 1 + √2) Use associative property of algebraic expressions. Associative property is written as. (a+b)+c=a+(b+c) This property modifies the expression as, 𝑃(𝑥) = ((𝑥 − 1) − √2) ((𝑥 − 1) + √2) Apply arithmetic rule. (𝑎 − 𝑏)(𝑎 + 𝑏) = 𝑎2 − 𝑏 2 Here, 𝑎 = (𝑥 − 1), 𝑏 = (√2) Hence, 𝑃(𝑥) = (𝑥 − 1)^{2} − (√22 Apply arithmetic rule: (𝑎 − 𝑏)2 = 𝑎2 − 2𝑎𝑏 + 𝑏 2 Here, a=x, b=1 Hence, 𝑃(𝑥) = 𝑥 2 − 2𝑥 + 1 − 2 = 𝑥 2 − 2𝑥 − 1 Hence, the product of [𝑥 − (1 + √2)][𝑥 − (1 − √2)] is 𝑥 2 − 2𝑥 − 1.