Answer :
Let's find the quotient by performing polynomial long division on [tex]\(8x^4 + 18x^3 - 48x^2 - 63x + 70\)[/tex] divided by [tex]\(2x^2 - 7\)[/tex].
### Step-by-Step Polynomial Long Division:
1. Setup:
[tex]\[
\text{Divide } 8x^4 + 18x^3 - 48x^2 - 63x + 70 \text{ by } 2x^2 - 7
\][/tex]
2. First Division:
- Divide the first term of the dividend [tex]\(8x^4\)[/tex] by the first term of the divisor [tex]\(2x^2\)[/tex]:
[tex]\[
\frac{8x^4}{2x^2} = 4x^2
\][/tex]
- Multiply the entire divisor by [tex]\(4x^2\)[/tex] and subtract from the original polynomial:
[tex]\[
(4x^2)(2x^2 - 7) = 8x^4 - 28x^2
\][/tex]
[tex]\[
(8x^4 + 18x^3 - 48x^2) - (8x^4 - 28x^2) = 18x^3 - 20x^2
\][/tex]
3. Second Division:
- Divide the first term of the new polynomial [tex]\(18x^3\)[/tex] by [tex]\(2x^2\)[/tex]:
[tex]\[
\frac{18x^3}{2x^2} = 9x
\][/tex]
- Multiply the divisor by [tex]\(9x\)[/tex] and subtract:
[tex]\[
(9x)(2x^2 - 7) = 18x^3 - 63x
\][/tex]
[tex]\[
(18x^3 - 20x^2 - 63x) - (18x^3 - 63x) = -20x^2 + 0x
\][/tex]
4. Third Division:
- Divide the new first term [tex]\(-20x^2\)[/tex] by [tex]\(2x^2\)[/tex]:
[tex]\[
\frac{-20x^2}{2x^2} = -10
\][/tex]
- Multiply the divisor by [tex]\(-10\)[/tex] and subtract:
[tex]\[
(-10)(2x^2 - 7) = -20x^2 + 70
\][/tex]
[tex]\[
(-20x^2 - 63x + 70) - (-20x^2 + 70) = -63x + 0
\][/tex]
5. Remainder:
- The remainder is [tex]\(-63x + 70\)[/tex].
### Result
The quotient of the division is:
[tex]\[
4x^2 + 9x - 10
\][/tex]
Thus, the answer is option C:
[tex]\[ 4x^2 + 9x - 10 \][/tex]
### Step-by-Step Polynomial Long Division:
1. Setup:
[tex]\[
\text{Divide } 8x^4 + 18x^3 - 48x^2 - 63x + 70 \text{ by } 2x^2 - 7
\][/tex]
2. First Division:
- Divide the first term of the dividend [tex]\(8x^4\)[/tex] by the first term of the divisor [tex]\(2x^2\)[/tex]:
[tex]\[
\frac{8x^4}{2x^2} = 4x^2
\][/tex]
- Multiply the entire divisor by [tex]\(4x^2\)[/tex] and subtract from the original polynomial:
[tex]\[
(4x^2)(2x^2 - 7) = 8x^4 - 28x^2
\][/tex]
[tex]\[
(8x^4 + 18x^3 - 48x^2) - (8x^4 - 28x^2) = 18x^3 - 20x^2
\][/tex]
3. Second Division:
- Divide the first term of the new polynomial [tex]\(18x^3\)[/tex] by [tex]\(2x^2\)[/tex]:
[tex]\[
\frac{18x^3}{2x^2} = 9x
\][/tex]
- Multiply the divisor by [tex]\(9x\)[/tex] and subtract:
[tex]\[
(9x)(2x^2 - 7) = 18x^3 - 63x
\][/tex]
[tex]\[
(18x^3 - 20x^2 - 63x) - (18x^3 - 63x) = -20x^2 + 0x
\][/tex]
4. Third Division:
- Divide the new first term [tex]\(-20x^2\)[/tex] by [tex]\(2x^2\)[/tex]:
[tex]\[
\frac{-20x^2}{2x^2} = -10
\][/tex]
- Multiply the divisor by [tex]\(-10\)[/tex] and subtract:
[tex]\[
(-10)(2x^2 - 7) = -20x^2 + 70
\][/tex]
[tex]\[
(-20x^2 - 63x + 70) - (-20x^2 + 70) = -63x + 0
\][/tex]
5. Remainder:
- The remainder is [tex]\(-63x + 70\)[/tex].
### Result
The quotient of the division is:
[tex]\[
4x^2 + 9x - 10
\][/tex]
Thus, the answer is option C:
[tex]\[ 4x^2 + 9x - 10 \][/tex]
