Answer :
Let's perform polynomial division to find the quotient when [tex]\(6x^4 + 16x^3 - 19x^2 - 20x + 19\)[/tex] is divided by [tex]\(-2x^2 + 3\)[/tex]. We'll also manage the remainder, if there is one, and express the result in the form [tex]\(q(x) + \frac{r(x)}{b(x)}\)[/tex].
### Step-by-Step Solution:
1. Setup Division:
Dividend: [tex]\(6x^4 + 16x^3 - 19x^2 - 20x + 19\)[/tex]
Divisor: [tex]\(-2x^2 + 3\)[/tex]
2. Start the Division Process:
- First Term: Divide the first term of the dividend by the first term of the divisor.
[tex]\[
\frac{6x^4}{-2x^2} = -3x^2
\][/tex]
- Multiply the entire divisor by [tex]\(-3x^2\)[/tex] and subtract from the dividend:
[tex]\[
(-3x^2)(-2x^2 + 3) = 6x^4 - 9x^2
\][/tex]
Subtract:
[tex]\[
(6x^4 + 16x^3 - 19x^2) - (6x^4 - 9x^2) = 16x^3 - 10x^2
\][/tex]
3. Repeat the Process:
- Second Term: Divide the first term of the new dividend by the first term of the divisor.
[tex]\[
\frac{16x^3}{-2x^2} = -8x
\][/tex]
- Multiply and subtract:
[tex]\[
(-8x)(-2x^2 + 3) = 16x^3 - 24x
\][/tex]
Subtract:
[tex]\[
(16x^3 - 10x^2 - 20x) - (16x^3 - 24x) = -10x^2 + 4x
\][/tex]
4. Continue the Process:
- Third Term: Divide the first term of the new dividend by the first term of the divisor:
[tex]\[
\frac{-10x^2}{-2x^2} = 5
\][/tex]
- Multiply and subtract:
[tex]\[
(5)(-2x^2 + 3) = -10x^2 + 15
\][/tex]
Subtract:
[tex]\[
(-10x^2 + 4x + 19) - (-10x^2 + 15) = 4x + 4
\][/tex]
5. Calculating Remainder:
- At this point, [tex]\(4x + 4\)[/tex] is of lower degree than the divisor [tex]\(-2x^2 + 3\)[/tex], so it's the remainder.
### Final Result:
The quotient [tex]\(q(x)\)[/tex] is [tex]\(-3x^2 - 8x + 5\)[/tex], and the remainder [tex]\(r(x)\)[/tex] is [tex]\(4x + 4\)[/tex].
Expressing the result in the requested form:
[tex]\[
q(x) + \frac{r(x)}{b(x)} = -3x^2 - 8x + 5 + \frac{4x + 4}{-2x^2 + 3}
\][/tex]
This gives us the complete solution to the division problem!
### Step-by-Step Solution:
1. Setup Division:
Dividend: [tex]\(6x^4 + 16x^3 - 19x^2 - 20x + 19\)[/tex]
Divisor: [tex]\(-2x^2 + 3\)[/tex]
2. Start the Division Process:
- First Term: Divide the first term of the dividend by the first term of the divisor.
[tex]\[
\frac{6x^4}{-2x^2} = -3x^2
\][/tex]
- Multiply the entire divisor by [tex]\(-3x^2\)[/tex] and subtract from the dividend:
[tex]\[
(-3x^2)(-2x^2 + 3) = 6x^4 - 9x^2
\][/tex]
Subtract:
[tex]\[
(6x^4 + 16x^3 - 19x^2) - (6x^4 - 9x^2) = 16x^3 - 10x^2
\][/tex]
3. Repeat the Process:
- Second Term: Divide the first term of the new dividend by the first term of the divisor.
[tex]\[
\frac{16x^3}{-2x^2} = -8x
\][/tex]
- Multiply and subtract:
[tex]\[
(-8x)(-2x^2 + 3) = 16x^3 - 24x
\][/tex]
Subtract:
[tex]\[
(16x^3 - 10x^2 - 20x) - (16x^3 - 24x) = -10x^2 + 4x
\][/tex]
4. Continue the Process:
- Third Term: Divide the first term of the new dividend by the first term of the divisor:
[tex]\[
\frac{-10x^2}{-2x^2} = 5
\][/tex]
- Multiply and subtract:
[tex]\[
(5)(-2x^2 + 3) = -10x^2 + 15
\][/tex]
Subtract:
[tex]\[
(-10x^2 + 4x + 19) - (-10x^2 + 15) = 4x + 4
\][/tex]
5. Calculating Remainder:
- At this point, [tex]\(4x + 4\)[/tex] is of lower degree than the divisor [tex]\(-2x^2 + 3\)[/tex], so it's the remainder.
### Final Result:
The quotient [tex]\(q(x)\)[/tex] is [tex]\(-3x^2 - 8x + 5\)[/tex], and the remainder [tex]\(r(x)\)[/tex] is [tex]\(4x + 4\)[/tex].
Expressing the result in the requested form:
[tex]\[
q(x) + \frac{r(x)}{b(x)} = -3x^2 - 8x + 5 + \frac{4x + 4}{-2x^2 + 3}
\][/tex]
This gives us the complete solution to the division problem!
