Math question algebra

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) = 9x³ − 4x² − 9x,    D(x) = 9x − 4

P(x) = ?

Answer 1

Assuming that Q(x) and R(x) are just other polynomials, you don't actually need polynomial division to get the answer that they're looking for. D(x) is already inside of P(x). Look what happens if you factor out x² from the first two terms of P(x):

P(x) = x²(9x - 4) - 9x,

= x²(D(x)) - 9x (Recall that D(x) = 9x-4))

= D(x) * x² + (-9x)

Q(x) = x², and your R(x) = -9x.

Here's what you get if you were to use standard polynomial long division:

x² - 1

9x - 4 | 9x³ - 4x² - 9x

-(9x³ - 4x²) ↓

,-9x

-(-9x + 4)

,-4

This shows that (9x^3 - 4x^2 - 9x)/(9x-4) = x^2 - 1 - 4/(9x-4). Multiplying both sides by 9x - 4:

9x^3 - 4x^2 - 9x = P(x) = (9x - 4)(x^2 - 1 - 4/(9x-4))

= x^2 * (9x-4) -(9x-4) - 4

= x^2 * (9x-4) - 9x

Which is the same answer we got from before.

Hope this helps!