Question

Divide the polynomial 12x^2+2x−9 by 4x−2 . Then express the polynomial as the product of the divisor and the quotient plus the remainder.

Answer 1

To divide the polynomial \( 12x^2 + 2x - 9 \) by \( 4x - 2 \), we can use polynomial long division.

1. **Set up the division**: We want to divide \( 12x^2 + 2x - 9 \) by \( 4x - 2 \).

2. **Divide the leading terms**: Divide the leading term of the dividend \( 12x^2 \) by the leading term of the divisor \( 4x \):
\[
\frac{12x^2}{4x} = 3x
\]

3. **Multiply the entire divisor by this result**: Multiply \( 3x \) by \( 4x - 2 \):
\[
3x(4x - 2) = 12x^2 - 6x
\]

4. **Subtract this from the original polynomial**:
\[
(12x^2 + 2x - 9) - (12x^2 - 6x) = 2x + 6x - 9 = 8x - 9
\]

5. **Repeat the process**: Now divide the leading term of the new polynomial \( 8x \) by the leading term of the divisor \( 4x \):
\[
\frac{8x}{4x} = 2
\]

6. **Multiply the entire divisor by this new result**: Multiply \( 2 \) by \( 4x - 2 \):
\[
2(4x - 2) = 8x - 4
\]

7. **Subtract this from the recent result**:
\[
(8x - 9) - (8x - 4) = -9 + 4 = -5
\]

At this point, we have reached the remainder of \(-5\).

### Final Result
Now we can express our original polynomial division as:
\[
12x^2 + 2x - 9 = (4x - 2)(3x + 2) - 5
\]

Thus, the polynomial \( 12x^2 + 2x - 9 \) divided by \( 4x - 2 \) yields a quotient of \( 3x + 2 \) and a remainder of \(-5\).