To find the quotient when dividing [tex]\(x^3 + 3x^2 - 16x - 48\)[/tex] by [tex]\(x + 3\)[/tex], we can perform polynomial long division or synthetic division.
Let's break down the process of polynomial long division step-by-step:
1. Set up the division:
We are dividing [tex]\(x^3 + 3x^2 - 16x - 48\)[/tex] by [tex]\(x + 3\)[/tex].
2. Divide the leading term:
- Divide the leading term of the dividend [tex]\(x^3\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex]:
[tex]\[
\frac{x^3}{x} = x^2
\][/tex]
3. Multiply and subtract:
- Multiply [tex]\(x^2\)[/tex] by [tex]\(x + 3\)[/tex] and subtract this product from the dividend.
[tex]\[
x^2 \cdot (x + 3) = x^3 + 3x^2
\][/tex]
- Subtract:
[tex]\[
(x^3 + 3x^2 - 16x - 48) - (x^3 + 3x^2) = -16x - 48
\][/tex]
4. Bring down the next term:
- Note that the next term is already considered, so we just perform the division on [tex]\(-16x - 48\)[/tex].
5. Repeat the process:
- Divide the leading term [tex]\(-16x\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{-16x}{x} = -16
\][/tex]
- Multiply [tex]\(-16\)[/tex] by [tex]\(x + 3\)[/tex]:
[tex]\[
-16 \cdot (x + 3) = -16x - 48
\][/tex]
- Subtract:
[tex]\[
(-16x - 48) - (-16x - 48) = 0
\][/tex]
So, the quotient is [tex]\(x^2 - 16\)[/tex] and the remainder is 0, meaning the division is exact.
Thus, [tex]\(x^3 + 3x^2 - 16x - 48\)[/tex] divided by [tex]\(x + 3\)[/tex] results in:
[tex]\[
\boxed{x^2 - 16}
\][/tex]
Therefore, the correct answer is:
B. [tex]\(x^2 - 16\)[/tex]
