Answer :
To divide the polynomial [tex]\(21x^4 + 85x^3 + 18x^2 - 11x - 23\)[/tex] by [tex]\(3x - 2\)[/tex], we will use polynomial long division. Here's a step-by-step solution:
1. Set up the division:
- Dividend: [tex]\(21x^4 + 85x^3 + 18x^2 - 11x - 23\)[/tex]
- Divisor: [tex]\(3x - 2\)[/tex]
2. Divide the leading terms:
- Divide the leading term of the dividend, [tex]\(21x^4\)[/tex], by the leading term of the divisor, [tex]\(3x\)[/tex]:
[tex]\[
\frac{21x^4}{3x} = 7x^3
\][/tex]
- This gives the first term of the quotient, [tex]\(7x^3\)[/tex].
3. Multiply and Subtract:
- Multiply [tex]\(7x^3\)[/tex] by the entire divisor [tex]\(3x - 2\)[/tex]:
[tex]\[
7x^3 \times (3x - 2) = 21x^4 - 14x^3
\][/tex]
- Subtract this result from the original dividend:
[tex]\[
(21x^4 + 85x^3 + 18x^2 - 11x - 23) - (21x^4 - 14x^3) = 99x^3 + 18x^2 - 11x - 23
\][/tex]
4. Repeat the process:
- Divide the new leading term [tex]\(99x^3\)[/tex] by [tex]\(3x\)[/tex]:
[tex]\[
\frac{99x^3}{3x} = 33x^2
\][/tex]
- Multiply [tex]\(33x^2\)[/tex] by the divisor:
[tex]\[
33x^2 \times (3x - 2) = 99x^3 - 66x^2
\][/tex]
- Subtract:
[tex]\[
(99x^3 + 18x^2 - 11x - 23) - (99x^3 - 66x^2) = 84x^2 - 11x - 23
\][/tex]
5. Continue dividing:
- Divide [tex]\(84x^2\)[/tex] by [tex]\(3x\)[/tex]:
[tex]\[
\frac{84x^2}{3x} = 28x
\][/tex]
- Multiply:
[tex]\[
28x \times (3x - 2) = 84x^2 - 56x
\][/tex]
- Subtract:
[tex]\[
(84x^2 - 11x - 23) - (84x^2 - 56x) = 45x - 23
\][/tex]
6. Final term:
- Divide [tex]\(45x\)[/tex] by [tex]\(3x\)[/tex]:
[tex]\[
\frac{45x}{3x} = 15
\][/tex]
- Multiply and subtract:
[tex]\[
15 \times (3x - 2) = 45x - 30
\][/tex]
[tex]\[
(45x - 23) - (45x - 30) = 7
\][/tex]
7. Result:
- The quotient is [tex]\(7x^3 + 33x^2 + 28x + 15\)[/tex].
- The remainder is [tex]\(7\)[/tex].
Therefore, the division of [tex]\(21x^4 + 85x^3 + 18x^2 - 11x - 23\)[/tex] by [tex]\(3x - 2\)[/tex] gives a quotient of [tex]\(7x^3 + 33x^2 + 28x + 15\)[/tex] with a remainder of 7.
1. Set up the division:
- Dividend: [tex]\(21x^4 + 85x^3 + 18x^2 - 11x - 23\)[/tex]
- Divisor: [tex]\(3x - 2\)[/tex]
2. Divide the leading terms:
- Divide the leading term of the dividend, [tex]\(21x^4\)[/tex], by the leading term of the divisor, [tex]\(3x\)[/tex]:
[tex]\[
\frac{21x^4}{3x} = 7x^3
\][/tex]
- This gives the first term of the quotient, [tex]\(7x^3\)[/tex].
3. Multiply and Subtract:
- Multiply [tex]\(7x^3\)[/tex] by the entire divisor [tex]\(3x - 2\)[/tex]:
[tex]\[
7x^3 \times (3x - 2) = 21x^4 - 14x^3
\][/tex]
- Subtract this result from the original dividend:
[tex]\[
(21x^4 + 85x^3 + 18x^2 - 11x - 23) - (21x^4 - 14x^3) = 99x^3 + 18x^2 - 11x - 23
\][/tex]
4. Repeat the process:
- Divide the new leading term [tex]\(99x^3\)[/tex] by [tex]\(3x\)[/tex]:
[tex]\[
\frac{99x^3}{3x} = 33x^2
\][/tex]
- Multiply [tex]\(33x^2\)[/tex] by the divisor:
[tex]\[
33x^2 \times (3x - 2) = 99x^3 - 66x^2
\][/tex]
- Subtract:
[tex]\[
(99x^3 + 18x^2 - 11x - 23) - (99x^3 - 66x^2) = 84x^2 - 11x - 23
\][/tex]
5. Continue dividing:
- Divide [tex]\(84x^2\)[/tex] by [tex]\(3x\)[/tex]:
[tex]\[
\frac{84x^2}{3x} = 28x
\][/tex]
- Multiply:
[tex]\[
28x \times (3x - 2) = 84x^2 - 56x
\][/tex]
- Subtract:
[tex]\[
(84x^2 - 11x - 23) - (84x^2 - 56x) = 45x - 23
\][/tex]
6. Final term:
- Divide [tex]\(45x\)[/tex] by [tex]\(3x\)[/tex]:
[tex]\[
\frac{45x}{3x} = 15
\][/tex]
- Multiply and subtract:
[tex]\[
15 \times (3x - 2) = 45x - 30
\][/tex]
[tex]\[
(45x - 23) - (45x - 30) = 7
\][/tex]
7. Result:
- The quotient is [tex]\(7x^3 + 33x^2 + 28x + 15\)[/tex].
- The remainder is [tex]\(7\)[/tex].
Therefore, the division of [tex]\(21x^4 + 85x^3 + 18x^2 - 11x - 23\)[/tex] by [tex]\(3x - 2\)[/tex] gives a quotient of [tex]\(7x^3 + 33x^2 + 28x + 15\)[/tex] with a remainder of 7.
