Answer :
To divide the polynomial [tex]\(3x^4 - 21x^2 + 20x + 6\)[/tex] by [tex]\(x + 3\)[/tex], you need to find the quotient using polynomial division. Here's how you can do it step-by-step:
1. Set up the division:
- Write [tex]\(3x^4 - 21x^2 + 20x + 6\)[/tex] under the long division bracket, and place [tex]\(x + 3\)[/tex] outside of it.
2. Divide the leading term:
- Look at the first term of the dividend, [tex]\(3x^4\)[/tex]. Divide this by the first term of the divisor, [tex]\(x\)[/tex].
- The result is [tex]\(3x^3\)[/tex].
3. Multiply and subtract:
- Multiply the entire divisor [tex]\(x + 3\)[/tex] by the result from step 2, which is [tex]\(3x^3\)[/tex], giving [tex]\(3x^4 + 9x^3\)[/tex].
- Subtract [tex]\(3x^4 + 9x^3\)[/tex] from the original polynomial:
[tex]\[
(3x^4 - 21x^2 + 20x + 6) - (3x^4 + 9x^3) = -9x^3 - 21x^2 + 20x + 6.
\][/tex]
4. Repeat the process:
- Divide [tex]\(-9x^3\)[/tex] by [tex]\(x\)[/tex], giving [tex]\(-9x^2\)[/tex].
- Multiply [tex]\(x + 3\)[/tex] by [tex]\(-9x^2\)[/tex], which gives [tex]\(-9x^3 - 27x^2\)[/tex].
- Subtract this from the current polynomial:
[tex]\[
(-9x^3 - 21x^2 + 20x + 6) - (-9x^3 - 27x^2) = 6x^2 + 20x + 6.
\][/tex]
5. Continue the division:
- Divide [tex]\(6x^2\)[/tex] by [tex]\(x\)[/tex], which results in [tex]\(6x\)[/tex].
- Multiply [tex]\(x + 3\)[/tex] by [tex]\(6x\)[/tex], giving [tex]\(6x^2 + 18x\)[/tex].
- Subtract to get:
[tex]\[
(6x^2 + 20x + 6) - (6x^2 + 18x) = 2x + 6.
\][/tex]
6. Final step:
- Divide [tex]\(2x\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(2\)[/tex].
- Multiply [tex]\(x + 3\)[/tex] by [tex]\(2\)[/tex], resulting in [tex]\(2x + 6\)[/tex].
- Subtract to find the remainder:
[tex]\[
(2x + 6) - (2x + 6) = 0.
\][/tex]
7. Summarize the result:
- Since the remainder is 0, the division is exact, and the quotient is [tex]\(3x^3 - 9x^2 + 6x + 2\)[/tex].
Therefore, when [tex]\(3x^4 - 21x^2 + 20x + 6\)[/tex] is divided by [tex]\(x+3\)[/tex], the quotient is [tex]\(3x^3 - 9x^2 + 6x + 2\)[/tex].
1. Set up the division:
- Write [tex]\(3x^4 - 21x^2 + 20x + 6\)[/tex] under the long division bracket, and place [tex]\(x + 3\)[/tex] outside of it.
2. Divide the leading term:
- Look at the first term of the dividend, [tex]\(3x^4\)[/tex]. Divide this by the first term of the divisor, [tex]\(x\)[/tex].
- The result is [tex]\(3x^3\)[/tex].
3. Multiply and subtract:
- Multiply the entire divisor [tex]\(x + 3\)[/tex] by the result from step 2, which is [tex]\(3x^3\)[/tex], giving [tex]\(3x^4 + 9x^3\)[/tex].
- Subtract [tex]\(3x^4 + 9x^3\)[/tex] from the original polynomial:
[tex]\[
(3x^4 - 21x^2 + 20x + 6) - (3x^4 + 9x^3) = -9x^3 - 21x^2 + 20x + 6.
\][/tex]
4. Repeat the process:
- Divide [tex]\(-9x^3\)[/tex] by [tex]\(x\)[/tex], giving [tex]\(-9x^2\)[/tex].
- Multiply [tex]\(x + 3\)[/tex] by [tex]\(-9x^2\)[/tex], which gives [tex]\(-9x^3 - 27x^2\)[/tex].
- Subtract this from the current polynomial:
[tex]\[
(-9x^3 - 21x^2 + 20x + 6) - (-9x^3 - 27x^2) = 6x^2 + 20x + 6.
\][/tex]
5. Continue the division:
- Divide [tex]\(6x^2\)[/tex] by [tex]\(x\)[/tex], which results in [tex]\(6x\)[/tex].
- Multiply [tex]\(x + 3\)[/tex] by [tex]\(6x\)[/tex], giving [tex]\(6x^2 + 18x\)[/tex].
- Subtract to get:
[tex]\[
(6x^2 + 20x + 6) - (6x^2 + 18x) = 2x + 6.
\][/tex]
6. Final step:
- Divide [tex]\(2x\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(2\)[/tex].
- Multiply [tex]\(x + 3\)[/tex] by [tex]\(2\)[/tex], resulting in [tex]\(2x + 6\)[/tex].
- Subtract to find the remainder:
[tex]\[
(2x + 6) - (2x + 6) = 0.
\][/tex]
7. Summarize the result:
- Since the remainder is 0, the division is exact, and the quotient is [tex]\(3x^3 - 9x^2 + 6x + 2\)[/tex].
Therefore, when [tex]\(3x^4 - 21x^2 + 20x + 6\)[/tex] is divided by [tex]\(x+3\)[/tex], the quotient is [tex]\(3x^3 - 9x^2 + 6x + 2\)[/tex].
