Introduction to Polynomials

Section P.3: Introduction to Polynomials Chapter P – Polynomials Definition of a monomial: A monomial is a variable, a real number, or a multiplication of one or more variables and a real number with whole-number exponents. A monomial may contain a numerical fraction, but it may not have a variable in the denominator of a fraction. #1-16: Classify the following terms as monomials or not monomials 1) 5x3 Solution: this is a monomial 3) 3x-4 Solution: this is not a monomial (because of the negative exponent) 5) -y6 Solution: this is a monomial 7) -11xy3 Solution: this is a monomial (it is okay to have more than one letter) 9) 2 𝑎𝑏 2 𝑐 3 Solution: this is a monomial (it is okay to have a number as a fraction, you just can’t have a letter in the denominator of a fraction) 11) 1 𝑦𝑥 −4 2 Solution: this is not a monomial because of the exponent of (-4) 13) 2 5𝑥 Solution: this is not a monomial (because of the x in the denominator of a fraction) 15) 2 𝑥+5 Solution: this is not a monomial (because of the x in the denominator of a fraction) #17 – 28: Determine the degree and coefficient of each of the following monomials 17) 8x2 Solution: this is a 2nd degree polynomial. The coefficient is the 8. 19) z3 I think of this as 1z3 Solution: This is a 3rd degree polynomial with coefficient of 1. 21) 2 𝑥 3 𝟐 Solution: This is a 1st degree polynomial with coefficient of 𝟑. 23) 3x2y6 I need to add the exponents of the x and the y to get the degree of the term. Solution: The degree is 8. The coefficient is 3. 25) -x4y5 Think of this as -1x4y5 I need to add the exponents of the x and y to get the degree. Solution: degree 9, coefficient -1 27) -xyz Think of this as -1x1y1z1 Add the exponents of the x, y, z to get the degree Solution: degree 3, coefficient -1 Polynomial definition: A polynomial is a monomial or the sum or difference of monomials. Each monomial is called a term of the polynomial. Important!:Terms are separated by addition signs and subtraction signs, but never by multiplication signs A polynomial with one term is called a monomial A polynomial with two terms is called a binomial A polynomial with three terms is called a trinomial #29-44: Classify the following terms as polynomials or not polynomials. If the expression is a polynomial classify it as a monomial, binomial, trinomial or other. 29) 4x + 5 Solution: this is a polynomial it is also a binomial 31) x2 + 5x +6 Solution: this is a polynomial it is also a trinomial 33) 3x-4 Solution: this is not a polynomial because of the negative exponent 35) x3 - 5 𝑥 Solution: this is not a polynomial because of the x in the denominator of a fraction 37) 𝑥+3 𝑥−4 Solution: this is not a polynomial because of the x in the denominator of a fraction 39) xy3 + 4x + 5y –7 Solution: this is a polynomial (but it is not a monomial / binomial nor trinomial) thus it is an other polynomial 41) 5x4 + 3xy Solution: this is a polynomial it is also a binomial 43) 5 + 3 𝑥+𝑦 Solution: this is not a polynomial because of the x and y in the denominator of a fraction The degree of a polynomial is the highest of the degrees of all its terms. The leading term of a polynomial is the term with the highest degree The leading coefficient of a polynomial is the coefficient of the term with the highest degree. #45-56: Find the leading term of each polynomial, then state the degree of the polynomial and the leading coefficient. 45) 5x3 + 3y Solution: the 5x3 is the leading term, the polynomial is degree 3, the leading coefficient is 5. 47) 3x3 + 5xyz3 The 3x3 term has degree 3, while the 5xyz3 term has degree 5 (add the exponents on each of the letters) Solution: the 5xyz3 is the leading term (because it has the highest degree), the polynomial is degree 5, the leading coefficient is 5 49) 8x + 5x2 – x4 The –x4 (think of this as -1x4) is the leading term as it has the degree 4 which is higher than the rest of the terms. Solution: -x4 is the leading term, the polynomial has degree 4, the leading coefficient is -1 51) -xy5 + 2yz – 3x2y If you add the exponents of the letters, the –xy5 (think of this as -1xy5) term has degree 6, which is the highest of all the terms. Solution: -xy5 is the leading term, the polynomial has degree 6, the leading coefficient is -1 53) 7 If I need to, I can think of this as 7x0 as 7x0 = 7*1 = 7 Solution: the 7 is the only term, thus it is the leading term. The polynomial is degree 0, the leading coefficient is 7. 55) −1 2 Solution: the coefficient is −1 is 2 −1 . 2 the only term, thus it is the leading term. The polynomial is degree 0, the leading #57 – 68: Evaluate each polynomial using x = 2, y = -3 and z = 4 57) 5x3 + 3y = 5(2)3 + 3(-3) = 5(8) + 3(-3) = 40 – 9 Solution 31 (it would be wrong to write x = 31, as x equals 2 in this problem) 59) 3x3 + 5xyz3 =3(2)3 + 5(2)(-3)(4)3 = 3(8) + 5(2)(-3)(64) = 24 – 1920 Solution: -1896 61) 8x + 5x2 – x4 8(2) + 5(2)2 – (2)4 8(2) + 5(4) – 16 16 + 20 – 16 Solution 20 63) -xy5 + 2yz – 3x2y I might think of this as -1xy5 + 2yz – 3x2y = -1(2)(-3)5 + 2(-3)(4) – 3(2)2(-3) = -1(2)(-243) + 2(-3)(4) – 3(4)(-3) = 486 – 24 + 36 Solution: 498 65) 7 Solution 7 (this is a constant function, since it doesn’t have any letters and it will always equal 7 regardless of the values of the letters) 67) −1 2 Solution −𝟏 𝟐 (this is a constant function, since it doesn’t have any letters and it will always equal 7 regardless of the values of the letters)