Polynomial division #2 - Questions and Answers

Question 1 The condition under which we may use synthetic division to divide polynomials Answer Given: Synthetic division of polynomial Key concepts applied: Synthetic division can be used whenever we need to divide a polynomial function by a binomial of the form x-c. By doing synthetic division, we can find the actual quotient and also the remainder we get when we divide the polynomial function by x-c. The Remainder Theorem states that the remainder that we end up with when synthetic division is applied actually gives us the functional value. Another use is finding factors and zeros. The Factor Theorem states that if the functional value is 0 at some value c, then x-c is a factor. Thus, we can not only find that functional value by using synthetic division, but also the quotient found can help with the factoring process. It is to be noted that, to divide polynomials using synthetic division, we must be dividing by a linear expression and the leading coefficient must be a 1. If the leading coefficient is not a 1, then we can perform synthetic division after dividing by the leading coefficient to turn the leading coefficient into a 1. Conclusion: To divide polynomials using synthetic division, we must be dividing by a linear expression and the leading coefficient must be a 1. If the leading coefficient is not a 1, then we can perform synthetic division after dividing by the leading coefficient to turn the leading coefficient into a 1. Question 2 The vertical, horizontal and oblique asymptotes of the rational function R(x) = 6z²+19z-7 / 3z-1 Answer Given: The rational function is defined by R(x) = 6z²+19z-7 / 3z-1 Result used: Vertical asymptote: Consider a rational function R(x) = p(x) / q(x), in its lowest terms. If r is a real zero of the polynomial q(x), then x = r is a vertical asymptote of the function R. Horizontal or oblique asymptotes: Consider a rational function R(x) = p(x) / q(x) = bmx^m + bm-1x^m-1+... b₁x + bo / anx^n+an-1x^n-1+... a₁x + ao where n is the degree of the polynomial p(x) and m is the degree of the polynomial q(x). If n = m + 1, the rational function R has no horizontal asymptote and only has an oblique asymptote given by y = ax + b where y = ax + b is quotient obtained by the polynomial division P(x) / q(x) Calculation: Rewrite the rational function R(x) = 6z²+19z-7 / 3z-1 as follows. R(x) = 6z²+19z-7 / 3z-1 = 6z²+21z-2z-7 / 3z-1 = 3z(2z+7)-1(2z+7) / 3z-1 = (2z+7)(3z-1) / 3z-1 = 2z +7 Hence, the rational function R(x) = 6z²+19z-7 / 3z-1 in its lowest terms is R(x) = 2z + 7. The polynomial in the denominator of R(x) = 2z + 7 is 1 and 1 has no zeros. Therefore, the rational function R(x) = 6z²+19z-7 / 3z-1 has no vertical asymptote. Note that, the degree of the polynomial in the numerator is 2 and the degree of the polynomial in the denominator is 1. That is, the degree of the polynomial in the numerator is 1 more than the degree of the polynomial in the denominator. Therefore, the rational function R(x) = 6z²+19z-7 / 3z-1 has no horizontal asymptote. The polynomial division 6z²+19z-7 / 3z-1 gives the quotient as 2z +7. Therefore, the rational function 6z²+19z-7 / 3z-1 has an oblique asymptote given by y = 2z + 7. Hence, the rational function 6z²+19z-7 / 3z-1 has no vertical asymptote, no horizontal ext and oblique asymptote is y = 2z. Question 3 Describe a use for the Remainder Theorem. Answer Concept used: Remainder theorem states that "If a polynomial f(x)is divided by x - k, then the remainder is the value f(k)" We can use polynomial division to evaluate polynomials using the Remainder Theorem. If the polynomial is divided by x - k, the remainder is r, then this value equals the value of the polynomial function at k, that is, f(k). So this helps to evaluate the polynomial at a given value of x. Since the division is done by a linear factor, we can use the synthetic division method.