Given the polynomial:
[tex]x^3-7x^2+7x+15[/tex]
You can factorize it as follow:
1. Rewrite the term with exponent 2 in this form:
[tex]-7x^2=x^2-8x^2[/tex]
2. Rewrite the x-term in this form:
[tex]7x=-8x+15x[/tex]
3. Rewrite the expression:
[tex]=x^3+x^2-8x^2-8x+15x+15[/tex]
4. Make three groups of two terms each using parentheses:
[tex]=(x^3+x^2)-(8x^2+8x)+(15x+15)[/tex]
5. Identify the Greatest Common Factor (GCF) of each group (the largest factor that all the terms in the group have in common):
- For:
[tex](x^3+x^2)[/tex]
The Greatest Common Factor is:
[tex]GCF=x^2[/tex]
- For:
[tex](8x^2+8x)[/tex]
The Greatest Common Factor is:
[tex]GCF=8x[/tex]
- And for:
[tex](15x+15)[/tex]
It is:
[tex]GCF=15[/tex]
6. Factor the GCF of each group out:
[tex]=x^2(x^{}+1)-8x(x+1)+15(x+1)[/tex]
7. Notice that each expression is common in all the terms:
[tex]x+1[/tex]
Then, you can factor it out:
[tex]=(x^{}+1)(x^2-8x+15)[/tex]
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:
[tex]=(x^{}+1)(x-3)(x-5)[/tex]
Hence, the answer is: Option D.
