Question

Rewrite (2x^6 - 9x^5 + 4x^2 - 5)/(x^3 - 5) in the form Q(x) + r(x)/b(x). What is Q(x)?

A) Q(x) = 2x^3 - 9x^2 + 4
B) Q(x) = 2x^3 - 9x^2 + 4x
C) Q(x) = 2x^4 - 9x^3 + 4
D) Q(x) = 2x^4 - 9x^3 + 4x

Answer 1

To rewrite the expression (2x^6 - 9x^5 + 4x^2 - 5)/(x^3 - 5) in the form Q(x) + r(x)/b(x), perform long division and simplify the result. The expression can be rewritten as Q(x) = 2x^3 - 32x^2 + 45.

Step-by-step explanation:

To rewrite the expression (2x^6 - 9x^5 + 4x^2 - 5)/(x^3 - 5) in the form Q(x) + r(x)/b(x), we need to perform long division. Here are the steps:

1. Divide 2x^6 by x^3 to get 2x^3.
2. Multiply (2x^3)(x^3 - 5) to get 2x^6 - 10x^3.
3. Subtract (2x^6 - 10x^3) from the original expression to get (-9x^5 + 4x^2 - 5 + 10x^3).
4. Divide -9x^5 by x^3 to get -9x^2.
5. Multiply (-9x^2)(x^3 - 5) to get -9x^5 + 45x^2.
6. Subtract (-9x^5 + 45x^2) from the previous result to get (4x^2 - 5 + 10x^3 + 9x^2 - 45x^2).
7. Combine like terms to simplify the expression to (10x^3 + 4x^2 - 40x^2 - 5).
8. Divide 10x^3 by x^3 to get 10.
9. Multiply (10)(x^3 - 5) to get 10x^3 - 50.
10. Subtract (10x^3 - 50) from the previous result to get (4x^2 - 5 + 10x^3 + 9x^2 - 45x^2 - 10x^3 + 50).
11. Combine like terms to simplify the expression to (13x^2 - 45x^2 + 50 - 5).
12. Simplify further to get (-32x^2 + 45).

Therefore, the expression (2x^6 - 9x^5 + 4x^2 - 5)/(x^3 - 5) can be rewritten in the form Q(x) + r(x)/b(x) as Q(x) = 2x^3 - 32x^2 + 45.