how do you To find the remainder when p(x) = -7x^2 + 2x + 40 is divided by D(x) = x - 5, and how do you solve to give a complete fractorization P(x) over the real numbers.

Question

I've been trying to figure it out and somewhat stuck on it.

Answer 1

Were you able to parse the response, above? It is a solid answer, but many of the items in it have been duplicated. If you can't work it out:

"The Remainder Theorem states that if a polynomial 𝑝(𝑥) is divided by (𝑥−𝑎), the remainder is 𝑝(𝑎)."

If p(x) = (-7)x^2 + (2)x + (40)

and

D(x) = x-5

then

(we remember that if "5" was a zero of p(x) then the remainder would = 0)

Remainder of p(x) / D(x) = p(5) = (-7)*(5)^2 + 2(5) + 40 = ... = -125.

Answer 2

To find the remainder of a polynomial division we will use the Remainder Theorem for the division.

Step 1: Finding the Remainder Using the Remainder Theorem

The Remainder Theorem states that if a polynomial 𝑝(𝑥)p(x) is divided by 𝑥−𝑎x−a, the remainder is 𝑝(𝑎)p(a).

Given the polynomials:

1. 𝑝(𝑥)=−7𝑥2+2𝑥+40p(x)=−7x2+2x+40
2. 𝐷(𝑥)=𝑥−5D(x)=x−5

To find the remainder when 𝑝(𝑥)p(x) is divided by 𝐷(𝑥)D(x), we substitute 𝑥=5x=5 into 𝑝(𝑥)p(x): 𝑝(5)=−7(5)2+2(5)+40p(5)=−7(5)2+2(5)+40 𝑝(5)=−7(25)+10+40p(5)=−7(25)+10+40 𝑝(5)=−175+10+40p(5)=−175+10+40 𝑝(5)=−125p(5)=−125

So, the remainder is −125.