To determine whether [tex]\((b + 4)\)[/tex] is a factor of the polynomial [tex]\(b^3 + 3b^2 - b + 12\)[/tex], we can use the Remainder Theorem and the Factor Theorem.
Here's a detailed, step-by-step solution:
1. Remainder Theorem:
- The Remainder Theorem states that the remainder of the division of a polynomial [tex]\(f(b)\)[/tex] by [tex]\((b - c)\)[/tex] is [tex]\(f(c)\)[/tex].
- In this case, we need to evaluate the polynomial at [tex]\(b = -4\)[/tex]. This means we substitute [tex]\(b = -4\)[/tex] into the polynomial [tex]\(b^3 + 3b^2 - b + 12\)[/tex].
2. Substitution:
- Let's substitute [tex]\(b = -4\)[/tex] into the polynomial:
[tex]\[
(-4)^3 + 3(-4)^2 - (-4) + 12
\][/tex]
3. Evaluate each term:
- [tex]\((-4)^3 = -64\)[/tex]
- [tex]\(3(-4)^2 = 3 \times 16 = 48\)[/tex]
- [tex]\(-(-4) = 4\)[/tex]
- [tex]\(12\)[/tex] remains as it is.
4. Combine the results:
- Now let's add these values together:
[tex]\[
-64 + 48 + 4 + 12
\][/tex]
[tex]\[
-64 + 48 = -16
\][/tex]
[tex]\[
-16 + 4 = -12
\][/tex]
[tex]\[
-12 + 12 = 0
\][/tex]
5. Conclusion based on the remainder:
- The resulting value is 0. According to the Factor Theorem, if substituting [tex]\(b\)[/tex] by a value [tex]\(c\)[/tex] into the polynomial results in 0, then [tex]\((b - c)\)[/tex] is a factor of the polynomial.
- In this case, substituting [tex]\(b = -4\)[/tex] results in 0, meaning [tex]\(b + 4\)[/tex] is indeed a factor of [tex]\(b^3 + 3b^2 - b + 12\)[/tex].
6. Select the correct option:
- Since the remainder is 0, and [tex]\((b + 4)\)[/tex] is a factor of the polynomial, the correct answer is:
[tex]\[
\boxed{A}
\][/tex]
