Question

Use long division to find the quotient of −6x3+17x2−32x+9/2x2−5x+9 . Enter the coefficients of the quotient.

Answer 1

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

1. **Set up the division**: Write \(-6x^3 + 17x^2 - 32x + 9\) under the long division symbol and \(2x^2 - 5x + 9\) outside.

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

3. **Multiply and subtract**: Multiply \(-3x\) by the entire divisor:
\[
-3x(2x^2 - 5x + 9) = -6x^3 + 15x^2 - 27x.
\]
Now, subtract this from the original polynomial:
\[
(-6x^3 + 17x^2 - 32x + 9) - (-6x^3 + 15x^2 - 27x) = (17x^2 - 15x^2) + (-32x + 27x) + 9 = 2x^2 - 5x + 9.
\]

4. **Repeat the process**: Now take the new polynomial \(2x^2 - 5x + 9\). Divide its leading term \(2x^2\) by the leading term of the divisor \(2x^2\):
\[
\frac{2x^2}{2x^2} = 1.
\]
Multiply \(1\) by the divisor:
\[
1(2x^2 - 5x + 9) = 2x^2 - 5x + 9.
\]
Subtract:
\[
(2x^2 - 5x + 9) - (2x^2 - 5x + 9) = 0.
\]

After completing the long division, the quotient is:
\[
-3x + 1
\]

Thus, the coefficients of the quotient are:
\[
-3, \quad 1
\]

The final answer for the coefficients of the quotient is \(-3, 1\).