Question

Use long division to find the quotient of −6x3+17x2−32x+92x2−5x+9
−
6
𝑥
3
+
17
𝑥
2
−
32
𝑥
+
9
2
𝑥
2
−
5
𝑥
+
9
. Enter the coefficients of the quotient.(1 point)


The quotient of −6x3+17x2−32x+92x2−5x+9
−
6
𝑥
3
+
17
𝑥
2
−
32
𝑥
+
9
2
𝑥
2
−
5
𝑥
+
9
is

Answer 1

To find the quotient of the polynomials \(-6x^3 + 17x^2 - 32x + 9\) divided by \(2x^2 - 5x + 9\) using long division, we can follow these steps:

1. **Arrange the polynomials in standard form**:
The dividend (the polynomial being divided) is
\[
-6x^3 + 17x^2 - 32x + 9
\]
and the divisor (the polynomial to divide by) is
\[
2x^2 - 5x + 9
\]

2. **Perform polynomial long division**:
- Divide the leading term of the dividend \(-6x^3\) by the leading term of the divisor \(2x^2\):
\[
\frac{-6x^3}{2x^2} = -3x
\]

- Multiply the entire divisor \(2x^2 - 5x + 9\) by \(-3x\):
\[
-3x(2x^2 - 5x + 9) = -6x^3 + 15x^2 - 27x
\]

- Subtract this from the original dividend:
\[
(-6x^3 + 17x^2 - 32x + 9) - (-6x^3 + 15x^2 - 27x) = (17x^2 - 15x^2) + (-32x + 27x) + 9
\]
\[
= 2x^2 - 5x + 9
\]

3. **Repeat the process**:
- Now divide the leading term \(2x^2\) by the leading term \(2x^2\):
\[
\frac{2x^2}{2x^2} = 1
\]

- Multiply the entire divisor by \(1\):
\[
1(2x^2 - 5x + 9) = 2x^2 - 5x + 9
\]

- Subtract this from the previous remainder:
\[
(2x^2 - 5x + 9) - (2x^2 - 5x + 9) = 0
\]

Since the remainder is \(0\), the process stops here.

4. **Write the result**:
The quotient is \(-3x + 1\).

Thus, the coefficients of the quotient \(-3x + 1\) are:
- For \(x^1\): \(-3\)
- For \(x^0\): \(1\)

Finally, the answer is:
\[
\boxed{[-3, 1]}
\]