Precalc 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

We can use synthetic division to divide P(x) by D(x). The root associated with D(x) = x - 4 is 4. Let's set up our synthetic division table:

4 | 1 3 0 -15 0

| 4 28 112 388

------------------------

1 7 28 97 388

(The second row of numbers is found by multiplying the root (4) by the value from the previous row, then adding it to the number above from the original polynomial. The process continues until all terms have been handled. We had to add a 0 as the coefficient of x^2 and as a constant term in P(x) since those terms were missing.)

The last number in the bottom row (388) is the remainder, and the rest of the numbers in that row give the coefficients of the quotient polynomial.

So, the quotient Q(x) is x^3 + 7x^2 + 28x + 97, and the remainder R(x) is 388.

Thus, we can express P(x) as:

P(x) = D(x) · Q(x) + R(x)

= (x - 4) · (x^3 + 7x^2 + 28x + 97) + 388