Answer :
To divide the polynomial [tex]\(-128h^3 - 62h - 78\)[/tex] by [tex]\(8h + 5\)[/tex], we will use polynomial long division. Here's a step-by-step explanation.
1. Setup the division problem:
Dividend: [tex]\(-128h^3 - 62h - 78\)[/tex]
Divisor: [tex]\(8h + 5\)[/tex]
2. Divide the first term:
Divide the first term of the dividend, [tex]\(-128h^3\)[/tex], by the first term of the divisor, [tex]\(8h\)[/tex]:
[tex]\[
\frac{-128h^3}{8h} = -16h^2
\][/tex]
3. Multiply and subtract:
Multiply the entire divisor [tex]\(8h + 5\)[/tex] by [tex]\(-16h^2\)[/tex]:
[tex]\[
(-16h^2) \cdot (8h + 5) = -128h^3 - 80h^2
\][/tex]
Subtract this result from the original dividend to find the new dividend:
[tex]\[
(-128h^3 - 62h - 78) - (-128h^3 - 80h^2) = 80h^2 - 62h - 78
\][/tex]
4. Repeat the process:
- Divide the first term of the new dividend [tex]\(80h^2\)[/tex] by the first term of the divisor [tex]\(8h\)[/tex]:
[tex]\[
\frac{80h^2}{8h} = 10h
\][/tex]
- Multiply the entire divisor [tex]\(8h + 5\)[/tex] by [tex]\(10h\)[/tex]:
[tex]\[
(10h) \cdot (8h + 5) = 80h^2 + 50h
\][/tex]
- Subtract this result from the current dividend:
[tex]\[
(80h^2 - 62h - 78) - (80h^2 + 50h) = -112h - 78
\][/tex]
5. Repeat the process again:
- Divide the first term of the new dividend [tex]\(-112h\)[/tex] by the first term of the divisor [tex]\(8h\)[/tex]:
[tex]\[
\frac{-112h}{8h} = -14
\][/tex]
- Multiply the entire divisor [tex]\(8h + 5\)[/tex] by [tex]\(-14\)[/tex]:
[tex]\[
(-14) \cdot (8h + 5) = -112h - 70
\][/tex]
- Subtract this result from the current dividend:
[tex]\[
(-112h - 78) - (-112h - 70) = -8
\][/tex]
6. Determine the remainder:
The final remainder is [tex]\(-8\)[/tex].
Since the polynomial does not divide evenly, we include the remainder as a fraction. The division result is:
[tex]\[
\text{Quotient: } -16h^2 + 10h - 14
\][/tex]
[tex]\[
\text{Remainder: } \frac{-8}{8h + 5}
\][/tex]
Therefore, the result of the division is:
[tex]\[
-16h^2 + 10h - 14 + \frac{-8}{8h + 5}
\][/tex]
1. Setup the division problem:
Dividend: [tex]\(-128h^3 - 62h - 78\)[/tex]
Divisor: [tex]\(8h + 5\)[/tex]
2. Divide the first term:
Divide the first term of the dividend, [tex]\(-128h^3\)[/tex], by the first term of the divisor, [tex]\(8h\)[/tex]:
[tex]\[
\frac{-128h^3}{8h} = -16h^2
\][/tex]
3. Multiply and subtract:
Multiply the entire divisor [tex]\(8h + 5\)[/tex] by [tex]\(-16h^2\)[/tex]:
[tex]\[
(-16h^2) \cdot (8h + 5) = -128h^3 - 80h^2
\][/tex]
Subtract this result from the original dividend to find the new dividend:
[tex]\[
(-128h^3 - 62h - 78) - (-128h^3 - 80h^2) = 80h^2 - 62h - 78
\][/tex]
4. Repeat the process:
- Divide the first term of the new dividend [tex]\(80h^2\)[/tex] by the first term of the divisor [tex]\(8h\)[/tex]:
[tex]\[
\frac{80h^2}{8h} = 10h
\][/tex]
- Multiply the entire divisor [tex]\(8h + 5\)[/tex] by [tex]\(10h\)[/tex]:
[tex]\[
(10h) \cdot (8h + 5) = 80h^2 + 50h
\][/tex]
- Subtract this result from the current dividend:
[tex]\[
(80h^2 - 62h - 78) - (80h^2 + 50h) = -112h - 78
\][/tex]
5. Repeat the process again:
- Divide the first term of the new dividend [tex]\(-112h\)[/tex] by the first term of the divisor [tex]\(8h\)[/tex]:
[tex]\[
\frac{-112h}{8h} = -14
\][/tex]
- Multiply the entire divisor [tex]\(8h + 5\)[/tex] by [tex]\(-14\)[/tex]:
[tex]\[
(-14) \cdot (8h + 5) = -112h - 70
\][/tex]
- Subtract this result from the current dividend:
[tex]\[
(-112h - 78) - (-112h - 70) = -8
\][/tex]
6. Determine the remainder:
The final remainder is [tex]\(-8\)[/tex].
Since the polynomial does not divide evenly, we include the remainder as a fraction. The division result is:
[tex]\[
\text{Quotient: } -16h^2 + 10h - 14
\][/tex]
[tex]\[
\text{Remainder: } \frac{-8}{8h + 5}
\][/tex]
Therefore, the result of the division is:
[tex]\[
-16h^2 + 10h - 14 + \frac{-8}{8h + 5}
\][/tex]
