Question

The reminder when the polynomial f(x) = 2x³ + px² + 9x + 18 is divided by (x-1) is 10 when it is divided by x 1 the reminder is 12 find the value of the constant p and q and the zeros of f (x)?

A) p=3,q=8,Zeros: x=−2,x=1
B) p=5,q=10,Zeros: x=−3,x=2
C) p=−4,q=7,Zeros: x=3,x=−2
D) p=2,q=6,Zeros: x=−1,x=2

Answer 1

Using the Remainder Theorem, we set up and solve equations by substituting x = 1 and x = -1 into the polynomial to find the constants p and q. Then we determine the zeros of the polynomial using factorization or the Rational Root Theorem.

Step-by-step explanation:

To solve for the value of the constant p and q, and find the zeros of the polynomial f(x) = 2x³ + px² + 9x + 18, we utilize the Remainder Theorem. This theorem states that if a polynomial f(x) is divided by (x - a), the remainder is f(a). Given the conditions that when f(x) is divided by (x - 1) the remainder is 10, and when divided by (x + 1) the remainder is 12, we substitute x = 1 and x = -1 into the polynomial respectively and set them equal to their remainders to create two equations.

We then solve these equations simultaneously to find the values of p and q. Lastly, to find the zeros of f(x), we would factor the polynomial or use the Rational Root Theorem and synthetic division to find the roots that satisfy the equation f(x) = 0.