College

Factor the polynomial: [tex]7x^8 - 56x^7 + 28x^6[/tex].

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. [tex]7x^8 - 56x^7 + 28x^6 =[/tex] [tex]\square[/tex]
(Type your answer in factored form. Simplify your answer.)

B. The polynomial is prime.

Answer :

To factor the polynomial [tex]\(7x^8 - 56x^7 + 28x^6\)[/tex], let's follow these steps:

1. Identify the Greatest Common Factor (GCF):
- Look at the coefficients: 7, 56, and 28. The greatest common factor of these numbers is 7.
- Look at the variable terms: [tex]\(x^8\)[/tex], [tex]\(x^7\)[/tex], and [tex]\(x^6\)[/tex]. The smallest power of [tex]\(x\)[/tex] is [tex]\(x^6\)[/tex].
- So, the GCF of the entire polynomial is [tex]\(7x^6\)[/tex].

2. Factor out the GCF:
- Divide each term in the polynomial by the GCF [tex]\(7x^6\)[/tex]:
- [tex]\(7x^8 \div 7x^6 = x^2\)[/tex]
- [tex]\(-56x^7 \div 7x^6 = -8x\)[/tex]
- [tex]\(28x^6 \div 7x^6 = 4\)[/tex]

3. Write the factored form:
- After factoring out the GCF, the expression becomes:
[tex]\[ 7x^6(x^2 - 8x + 4) \][/tex]

The factored form of the polynomial [tex]\(7x^8 - 56x^7 + 28x^6\)[/tex] is:
[tex]\[ 7x^6(x^2 - 8x + 4) \][/tex]

This gives us the complete factored form.