To find the quotient of [tex]\(x^2 + 7x + 12\)[/tex] divided by [tex]\(x + 4\)[/tex], we can use polynomial long division. Here’s a detailed, step-by-step solution:
1. Setup the Division:
- Write the dividend ([tex]\(x^2 + 7x + 12\)[/tex]) inside the division symbol.
- Write the divisor ([tex]\(x + 4\)[/tex]) outside the division symbol.
2. Divide the Leading Terms:
- Divide the leading term of the dividend ([tex]\(x^2\)[/tex]) by the leading term of the divisor ([tex]\(x\)[/tex]).
- [tex]\(x^2 / x = x\)[/tex].
3. Multiply and Subtract:
- Multiply the entire divisor by [tex]\(x\)[/tex] (the result of the previous step):
[tex]\[(x + 4) \times x = x^2 + 4x\][/tex].
- Subtract this product from the original polynomial:
[tex]\[
(x^2 + 7x + 12) - (x^2 + 4x) = 3x + 12
\][/tex].
- Now, the new polynomial is [tex]\(3x + 12\)[/tex].
4. Repeat the Process:
- Divide the leading term of the new polynomial ([tex]\(3x\)[/tex]) by the leading term of the divisor ([tex]\(x\)[/tex]):
[tex]\[
3x / x = 3
\][/tex].
- Multiply the entire divisor by [tex]\(3\)[/tex]:
[tex]\[
(x + 4) \times 3 = 3x + 12
\][/tex].
- Subtract this product from the new polynomial:
[tex]\[
(3x + 12) - (3x + 12) = 0
\][/tex].
- Now, the remainder is [tex]\(0\)[/tex].
5. Combine the Results:
- The quotient, obtained from the steps above, is [tex]\(x + 3\)[/tex].
- The remainder is [tex]\(0\)[/tex].
Conclusion:
The quotient of [tex]\(x^2 + 7x + 12\)[/tex] divided by [tex]\(x + 4\)[/tex] is [tex]\(x + 3\)[/tex], and the remainder is [tex]\(0\)[/tex]. Hence,
[tex]\[ \frac{x^2 + 7x + 12}{x + 4} = x + 3 \][/tex].
