Certainly! To solve the division of the polynomial [tex]\(x^4 + 11x^3 + 33x^2 + 24x + 32\)[/tex] by [tex]\(x + 6\)[/tex], we will use polynomial long division. Here are the steps in detail:
1. Set Up the Division:
[tex]\[
\frac{x^4 + 11x^3 + 33x^2 + 24x + 32}{x + 6}
\][/tex]
2. Divide the Leading Terms:
Divide the leading term of the dividend (which is [tex]\(x^4\)[/tex]) by the leading term of the divisor (which is [tex]\(x\)[/tex]):
[tex]\[
x^4 \div x = x^3
\][/tex]
This gives us the first term of the quotient: [tex]\(x^3\)[/tex].
3. Multiply and Subtract:
Multiply [tex]\(x^3\)[/tex] by [tex]\(x + 6\)[/tex], and subtract the result from the dividend:
[tex]\[
x^3 \cdot (x + 6) = x^4 + 6x^3
\][/tex]
Subtract this product from the original polynomial:
[tex]\[
(x^4 + 11x^3 + 33x^2 + 24x + 32) - (x^4 + 6x^3) = 5x^3 + 33x^2 + 24x + 32
\][/tex]
4. Repeat the Process:
Repeat the division with the new polynomial [tex]\(5x^3 + 33x^2 + 24x + 32\)[/tex].
- Divide the leading term [tex]\(5x^3\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
5x^3 \div x = 5x^2
\][/tex]
This gives the next term of the quotient: [tex]\(5x^2\)[/tex].
- Multiply and subtract:
[tex]\[
5x^2 \cdot (x + 6) = 5x^3 + 30x^2
\][/tex]
Subtract this result from the new polynomial:
[tex]\[
(5x^3 + 33x^2 + 24x + 32) - (5x^3 + 30x^2) = 3x^2 + 24x + 32
\][/tex]
5. Continue the Procedure:
- Divide the leading term [tex]\(3x^2\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
3x^2 \div x = 3x
\][/tex]
This gives us the next term of the quotient: [tex]\(3x\)[/tex].
- Multiply and subtract:
[tex]\[
3x \cdot (x + 6) = 3x^2 + 18x
\][/tex]
Subtract this result from the current polynomial:
[tex]\[
(3x^2 + 24x + 32) - (3x^2 + 18x) = 6x + 32
\][/tex]
6. Final Steps:
- Divide the leading term [tex]\(6x\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
6x \div x = 6
\][/tex]
This gives the last term of the quotient: [tex]\(6\)[/tex].
- Multiply and subtract:
[tex]\[
6 \cdot (x + 6) = 6x + 36
\][/tex]
Subtract this from the current polynomial:
[tex]\[
(6x + 32) - (6x + 36) = -4
\][/tex]
The remainder is [tex]\(-4\)[/tex].
So, the quotient is:
[tex]\[
x^3 + 5x^2 + 3x + 6
\][/tex]
and the remainder is:
[tex]\[
-4
\][/tex]
Therefore, the division can be written as:
[tex]\[
\frac{x^4 + 11x^3 + 33x^2 + 24x + 32}{x + 6} = x^3 + 5x^2 + 3x + 6 - \frac{4}{x + 6}
\][/tex]
