Answer :
To solve the polynomial division problem [tex]\((4x^3 + 19x^2 + 27x - 8) \div (4x - 1)\)[/tex], we will perform polynomial long division. Here's a step-by-step explanation:
1. Divide the first term:
- Take the leading term of the dividend, [tex]\(4x^3\)[/tex], and divide it by the leading term of the divisor, [tex]\(4x\)[/tex].
- This yields [tex]\(x^2\)[/tex].
2. Multiply and subtract:
- Multiply the entire divisor, [tex]\(4x - 1\)[/tex], by [tex]\(x^2\)[/tex]:
[tex]\[
x^2 \cdot (4x - 1) = 4x^3 - x^2
\][/tex]
- Subtract this result from the original polynomial:
[tex]\[
(4x^3 + 19x^2 + 27x - 8) - (4x^3 - x^2) = 20x^2 + 27x - 8
\][/tex]
3. Repeat the process:
- Divide the new leading term, [tex]\(20x^2\)[/tex], by [tex]\(4x\)[/tex] to get [tex]\(5x\)[/tex].
4. Multiply and subtract:
- Multiply the entire divisor by [tex]\(5x\)[/tex]:
[tex]\[
5x \cdot (4x - 1) = 20x^2 - 5x
\][/tex]
- Subtract this from the result of the previous subtraction:
[tex]\[
(20x^2 + 27x - 8) - (20x^2 - 5x) = 32x - 8
\][/tex]
5. Repeat the process again:
- Divide the leading term, [tex]\(32x\)[/tex], by [tex]\(4x\)[/tex] to get [tex]\(8\)[/tex].
6. Multiply and subtract:
- Multiply the entire divisor by [tex]\(8\)[/tex]:
[tex]\[
8 \cdot (4x - 1) = 32x - 8
\][/tex]
- Subtract this from the last remainder:
[tex]\[
(32x - 8) - (32x - 8) = 0
\][/tex]
Since the remainder is 0, the division is exact, and the quotient of the division is [tex]\(x^2 + 5x + 8\)[/tex].
1. Divide the first term:
- Take the leading term of the dividend, [tex]\(4x^3\)[/tex], and divide it by the leading term of the divisor, [tex]\(4x\)[/tex].
- This yields [tex]\(x^2\)[/tex].
2. Multiply and subtract:
- Multiply the entire divisor, [tex]\(4x - 1\)[/tex], by [tex]\(x^2\)[/tex]:
[tex]\[
x^2 \cdot (4x - 1) = 4x^3 - x^2
\][/tex]
- Subtract this result from the original polynomial:
[tex]\[
(4x^3 + 19x^2 + 27x - 8) - (4x^3 - x^2) = 20x^2 + 27x - 8
\][/tex]
3. Repeat the process:
- Divide the new leading term, [tex]\(20x^2\)[/tex], by [tex]\(4x\)[/tex] to get [tex]\(5x\)[/tex].
4. Multiply and subtract:
- Multiply the entire divisor by [tex]\(5x\)[/tex]:
[tex]\[
5x \cdot (4x - 1) = 20x^2 - 5x
\][/tex]
- Subtract this from the result of the previous subtraction:
[tex]\[
(20x^2 + 27x - 8) - (20x^2 - 5x) = 32x - 8
\][/tex]
5. Repeat the process again:
- Divide the leading term, [tex]\(32x\)[/tex], by [tex]\(4x\)[/tex] to get [tex]\(8\)[/tex].
6. Multiply and subtract:
- Multiply the entire divisor by [tex]\(8\)[/tex]:
[tex]\[
8 \cdot (4x - 1) = 32x - 8
\][/tex]
- Subtract this from the last remainder:
[tex]\[
(32x - 8) - (32x - 8) = 0
\][/tex]
Since the remainder is 0, the division is exact, and the quotient of the division is [tex]\(x^2 + 5x + 8\)[/tex].
