Answer :
To solve the problem of finding the remaining factors of the polynomial [tex]\(x^3 + x^2 - 16x - 16\)[/tex], given that [tex]\(x+1\)[/tex] is a factor, follow these steps:
1. Identify the Polynomial: We are given the polynomial [tex]\(x^3 + x^2 - 16x - 16\)[/tex] and the factor [tex]\(x + 1\)[/tex].
2. Use Polynomial Division: Since [tex]\(x+1\)[/tex] is a factor, we can perform polynomial division of [tex]\(x^3 + x^2 - 16x - 16\)[/tex] by [tex]\(x + 1\)[/tex] to find another factor. When dividing, the remainder should be 0, confirming [tex]\(x + 1\)[/tex] is a factor. The quotient we get is another factor of the polynomial.
3. Calculate the Quotient: The result of the division is [tex]\(x^2 - 16\)[/tex], meaning that the polynomial [tex]\(x^3 + x^2 - 16x - 16\)[/tex] can be expressed as [tex]\((x + 1)(x^2 - 16)\)[/tex].
4. Factor the Quotient Further: We now need to further factor the quotient [tex]\(x^2 - 16\)[/tex]. Recognize that this is a difference of squares:
[tex]\[
x^2 - 16 = (x - 4)(x + 4)
\][/tex]
5. List All Factors: Therefore, the complete factorization of the original polynomial is:
[tex]\[
(x + 1)(x - 4)(x + 4)
\][/tex]
Based on this factorization, the remaining factors of the polynomial [tex]\((x^3 + x^2 - 16x - 16)\)[/tex], after confirming [tex]\(x+1\)[/tex] is a factor, are [tex]\(x - 4\)[/tex] and [tex]\(x + 4\)[/tex].
So, the factors from the given options that apply are:
- D. [tex]\(x-4\)[/tex]
- E. [tex]\(x+4\)[/tex]
1. Identify the Polynomial: We are given the polynomial [tex]\(x^3 + x^2 - 16x - 16\)[/tex] and the factor [tex]\(x + 1\)[/tex].
2. Use Polynomial Division: Since [tex]\(x+1\)[/tex] is a factor, we can perform polynomial division of [tex]\(x^3 + x^2 - 16x - 16\)[/tex] by [tex]\(x + 1\)[/tex] to find another factor. When dividing, the remainder should be 0, confirming [tex]\(x + 1\)[/tex] is a factor. The quotient we get is another factor of the polynomial.
3. Calculate the Quotient: The result of the division is [tex]\(x^2 - 16\)[/tex], meaning that the polynomial [tex]\(x^3 + x^2 - 16x - 16\)[/tex] can be expressed as [tex]\((x + 1)(x^2 - 16)\)[/tex].
4. Factor the Quotient Further: We now need to further factor the quotient [tex]\(x^2 - 16\)[/tex]. Recognize that this is a difference of squares:
[tex]\[
x^2 - 16 = (x - 4)(x + 4)
\][/tex]
5. List All Factors: Therefore, the complete factorization of the original polynomial is:
[tex]\[
(x + 1)(x - 4)(x + 4)
\][/tex]
Based on this factorization, the remaining factors of the polynomial [tex]\((x^3 + x^2 - 16x - 16)\)[/tex], after confirming [tex]\(x+1\)[/tex] is a factor, are [tex]\(x - 4\)[/tex] and [tex]\(x + 4\)[/tex].
So, the factors from the given options that apply are:
- D. [tex]\(x-4\)[/tex]
- E. [tex]\(x+4\)[/tex]
