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:
-
Set up the division: Write \(-6x^3 + 17x^2 - 32x + 9\) under the long division symbol and \(2x^2 - 5x + 9\) outside.
-
Divide the leading term: Divide the leading term of the dividend \(-6x^3\) by the leading term of the divisor \(2x^2\): \[ \frac{-6x^3}{2x^2} = -3x. \]
-
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: \[ (-6x^3 + 17x^2 - 32x + 9) - (-6x^3 + 15x^2 - 27x) = (17x^2 - 15x^2) + (-32x + 27x) + 9 = 2x^2 - 5x + 9. \]
-
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 \(-3, 1\).