Answer :
Let's solve the problem: [tex]\((6x^4 + 16x^3 - 19x^2 - 20x + 19) \div (-2x^2 + 3)\)[/tex].
### Step-by-Step Division
1. Arrange the polynomials:
The dividend is [tex]\(6x^4 + 16x^3 - 19x^2 - 20x + 19\)[/tex], and the divisor is [tex]\(-2x^2 + 3\)[/tex].
2. Divide the leading term of the dividend by the leading term of the divisor:
- [tex]\(\frac{6x^4}{-2x^2} = -3x^2\)[/tex].
3. Multiply the entire divisor by [tex]\(-3x^2\)[/tex] and subtract from the dividend:
- [tex]\((-3x^2) \times (-2x^2 + 3) = 6x^4 - 9x^2\)[/tex].
- Subtract this from the original dividend:
[tex]\[
(6x^4 + 16x^3 - 19x^2 - 20x + 19) - (6x^4 - 9x^2) = 16x^3 - 10x^2 - 20x + 19.
\][/tex]
4. Repeat the process with the new polynomial:
- Divide [tex]\(16x^3\)[/tex] by [tex]\(-2x^2\)[/tex]: [tex]\(\frac{16x^3}{-2x^2} = -8x\)[/tex].
- Multiply the divisor by [tex]\(-8x\)[/tex] and subtract:
[tex]\[
(-8x) \times (-2x^2 + 3) = 16x^3 - 24x.
\][/tex]
- Subtract:
[tex]\[
(16x^3 - 10x^2 - 20x + 19) - (16x^3 - 24x) = -10x^2 + 4x + 19.
\][/tex]
5. Continue with [tex]\(-10x^2 + 4x + 19\)[/tex]:
- Divide [tex]\(-10x^2\)[/tex] by [tex]\(-2x^2\)[/tex]: [tex]\(\frac{-10x^2}{-2x^2} = 5\)[/tex].
- Multiply the divisor by 5 and subtract:
[tex]\[
5 \times (-2x^2 + 3) = -10x^2 + 15.
\][/tex]
- Subtract:
[tex]\[
(-10x^2 + 4x + 19) - (-10x^2 + 15) = 4x + 4.
\][/tex]
Now, we have completed the division process. The quotient is [tex]\(-3x^2 - 8x + 5\)[/tex] and the remainder is [tex]\(4x + 4\)[/tex].
### Final Result
The result of the division is:
[tex]\[
\left(6x^4 + 16x^3 - 19x^2 - 20x + 19\right) \div \left(-2x^2 + 3\right) = -3x^2 - 8x + 5 \quad \text{with a remainder of} \quad 4x + 4.
\][/tex]
So, the solution can be expressed as:
[tex]\[
-3x^2 - 8x + 5 + \frac{4x + 4}{-2x^2 + 3}
\][/tex]
### Step-by-Step Division
1. Arrange the polynomials:
The dividend is [tex]\(6x^4 + 16x^3 - 19x^2 - 20x + 19\)[/tex], and the divisor is [tex]\(-2x^2 + 3\)[/tex].
2. Divide the leading term of the dividend by the leading term of the divisor:
- [tex]\(\frac{6x^4}{-2x^2} = -3x^2\)[/tex].
3. Multiply the entire divisor by [tex]\(-3x^2\)[/tex] and subtract from the dividend:
- [tex]\((-3x^2) \times (-2x^2 + 3) = 6x^4 - 9x^2\)[/tex].
- Subtract this from the original dividend:
[tex]\[
(6x^4 + 16x^3 - 19x^2 - 20x + 19) - (6x^4 - 9x^2) = 16x^3 - 10x^2 - 20x + 19.
\][/tex]
4. Repeat the process with the new polynomial:
- Divide [tex]\(16x^3\)[/tex] by [tex]\(-2x^2\)[/tex]: [tex]\(\frac{16x^3}{-2x^2} = -8x\)[/tex].
- Multiply the divisor by [tex]\(-8x\)[/tex] and subtract:
[tex]\[
(-8x) \times (-2x^2 + 3) = 16x^3 - 24x.
\][/tex]
- Subtract:
[tex]\[
(16x^3 - 10x^2 - 20x + 19) - (16x^3 - 24x) = -10x^2 + 4x + 19.
\][/tex]
5. Continue with [tex]\(-10x^2 + 4x + 19\)[/tex]:
- Divide [tex]\(-10x^2\)[/tex] by [tex]\(-2x^2\)[/tex]: [tex]\(\frac{-10x^2}{-2x^2} = 5\)[/tex].
- Multiply the divisor by 5 and subtract:
[tex]\[
5 \times (-2x^2 + 3) = -10x^2 + 15.
\][/tex]
- Subtract:
[tex]\[
(-10x^2 + 4x + 19) - (-10x^2 + 15) = 4x + 4.
\][/tex]
Now, we have completed the division process. The quotient is [tex]\(-3x^2 - 8x + 5\)[/tex] and the remainder is [tex]\(4x + 4\)[/tex].
### Final Result
The result of the division is:
[tex]\[
\left(6x^4 + 16x^3 - 19x^2 - 20x + 19\right) \div \left(-2x^2 + 3\right) = -3x^2 - 8x + 5 \quad \text{with a remainder of} \quad 4x + 4.
\][/tex]
So, the solution can be expressed as:
[tex]\[
-3x^2 - 8x + 5 + \frac{4x + 4}{-2x^2 + 3}
\][/tex]
