Answer :
To solve the problem, we need to evaluate the factorizations made by Will and Olivia for the expression [tex]\(7x^6\)[/tex].
First, let's examine Will's factorization:
1. Will factored [tex]\(7x^6\)[/tex] as [tex]\((3x^2)(4x^4)\)[/tex].
- When we expand [tex]\((3x^2)(4x^4)\)[/tex], we multiply the coefficients and the powers of [tex]\(x\)[/tex] separately:
[tex]\[
3 \times 4 = 12 \quad \text{(coefficients)}
\][/tex]
[tex]\[
x^2 \times x^4 = x^{2+4} = x^6 \quad \text{(sum of exponents)}
\][/tex]
- So, Will's factorization gives us [tex]\(12x^6\)[/tex], which is not the same as the original expression [tex]\(7x^6\)[/tex].
Now, let's evaluate Olivia's factorization:
2. Olivia factored [tex]\(7x^6\)[/tex] as [tex]\((7x^2)(x^3)\)[/tex].
- When we expand [tex]\((7x^2)(x^3)\)[/tex], we again multiply the coefficients and the powers of [tex]\(x\)[/tex]:
[tex]\[
7 \times 1 = 7 \quad \text{(coefficients)}
\][/tex]
[tex]\[
x^2 \times x^3 = x^{2+3} = x^5 \quad \text{(sum of exponents)}
\][/tex]
- So, Olivia's factorization gives us [tex]\(7x^5\)[/tex], which is also not the same as the original expression [tex]\(7x^6\)[/tex].
Based on this analysis, neither Will nor Olivia factored [tex]\(7x^6\)[/tex] correctly. Therefore, the correct answer is:
(D) Neither Will nor Olivia
First, let's examine Will's factorization:
1. Will factored [tex]\(7x^6\)[/tex] as [tex]\((3x^2)(4x^4)\)[/tex].
- When we expand [tex]\((3x^2)(4x^4)\)[/tex], we multiply the coefficients and the powers of [tex]\(x\)[/tex] separately:
[tex]\[
3 \times 4 = 12 \quad \text{(coefficients)}
\][/tex]
[tex]\[
x^2 \times x^4 = x^{2+4} = x^6 \quad \text{(sum of exponents)}
\][/tex]
- So, Will's factorization gives us [tex]\(12x^6\)[/tex], which is not the same as the original expression [tex]\(7x^6\)[/tex].
Now, let's evaluate Olivia's factorization:
2. Olivia factored [tex]\(7x^6\)[/tex] as [tex]\((7x^2)(x^3)\)[/tex].
- When we expand [tex]\((7x^2)(x^3)\)[/tex], we again multiply the coefficients and the powers of [tex]\(x\)[/tex]:
[tex]\[
7 \times 1 = 7 \quad \text{(coefficients)}
\][/tex]
[tex]\[
x^2 \times x^3 = x^{2+3} = x^5 \quad \text{(sum of exponents)}
\][/tex]
- So, Olivia's factorization gives us [tex]\(7x^5\)[/tex], which is also not the same as the original expression [tex]\(7x^6\)[/tex].
Based on this analysis, neither Will nor Olivia factored [tex]\(7x^6\)[/tex] correctly. Therefore, the correct answer is:
(D) Neither Will nor Olivia
