Precalculus Polynomials

Question

Two polynomials P and D are given. Use either synthetic or long division to divide P(x)

 by D(x),

 and express P in the form P(x) = D(x) · Q(x) + R(x).

P(x) = x⁴ + 3x³ − 15x,    D(x) = x − 4

P(x)=

Answer 1

This problem requires dividing P(x), (x⁴ + 3x³ − 15x), by D(x), , and then writing the P(x) function as a product and sum of the divisor and quotient and remainder.

Start by synthetically dividing P(x) by D(x).

4 | 1 3 0 -15 0

| 4 28 112 338

` -----------------------------

1 7 28 97 |338

The resulting quotient, Q(x), is (x³ + 7x² + 28x + 97), with the remainder, R(x) of 338.

Expressing P(x) as a product and sum, P(x) = Q(x)*D(x) + R(X).

Therefore, P(x) = (x³ + 7x² + 28x + 97)*(x - 4) + 338.

To check, distribute the polynomials and add the remainder, to return x⁴ + 3x³ − 15x.