Question 6 (2x^3-21x^2-4x+16)/(2x-1)

To solve the given expression, we need to perform polynomial long division. Let's divide the numerator by the denominator step by step:<br /><br />1. Divide the leading term of the numerator (2x^3) by the leading term of the denominator (2x). This gives us x^2.<br />2. Multiply x^2 by the entire denominator (2x - 1) to get 2x^3 - x^2.<br />3. Subtract this result from the numerator to get -21x^2 - 4x + 16.<br />4. Repeat steps 1-3 with the new numerator. Divide -21x^2 by 2x to get -10.5x.<br />5. Multiply -10.5x by the entire denominator to get -21x^2 + 10.5x.<br />6. Subtract this result from the new numerator to get -14.5x + 16.<br />7. Repeat steps 1-3 with the new numerator. Divide -14.5x by 2x to get -7.25.<br />8. Multiply -7.25 by the entire denominator to get -14.5x + 7.25.<br />9. Subtract this result from the new numerator to get 8.75.<br /><br />Therefore, the simplified expression is:<br /><br />$\frac {2x^{3}-21x^{2}-4x+16}{2x-1} = x^2 - 10.5x - 7.25 + \frac{8.75}{2x-1}$