Question

Which of the following is the complete factorization of the polynomial below? x3 -7% +7x+15 O A. (x + 1)(x+3)(x+5) O B. (x-1)(x-3)(x-5) (C. (x-1)(x+3)(x+5) OD. (16-3 D. (x+1(x-3)(x-5)

Answer 1

Given the polynomial:

`x^3-7x^2+7x+15`

You can factorize it as follow:

1. Rewrite the term with exponent 2 in this form:

`-7x^2=x^2-8x^2`

2. Rewrite the x-term in this form:

`7x=-8x+15x`

3. Rewrite the expression:

`=x^3+x^2-8x^2-8x+15x+15`

4. Make three groups of two terms each using parentheses:

`=(x^3+x^2)-(8x^2+8x)+(15x+15)`

5. Identify the Greatest Common Factor (GCF) of each group (the largest factor that all the terms in the group have in common):

- For:

`(x^3+x^2)`

The Greatest Common Factor is:

`GCF=x^2`

- For:

`(8x^2+8x)`

The Greatest Common Factor is:

`GCF=8x`

- And for:

`(15x+15)`

It is:

`GCF=15`

6. Factor the GCF of each group out:

`=x^2(x^{}+1)-8x(x+1)+15(x+1)`

7. Notice that each expression is common in all the terms:

`x+1`

Then, you can factor it out:

`=(x^{}+1)(x^2-8x+15)`

8. In order to factor the Quadratic Polynomial in the second parentheses, you can find two numbers whose Sum is -8 and whose Product is 15. These are -3 and -5. Then, you get:

`=(x^{}+1)(x-3)(x-5)`

Hence, the answer is: Option D.