Question

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Question
Divide the polynomial 12x2+2x−9
by 4x−2
. Then express the polynomial as the product of the divisor and the quotient plus the remainder. (1 point)
Responses

12x2+2x−9=(4x−2)(3x+2)+5
12 x squared plus 2 x minus 9 is equal to open paren 4 x minus 2 close paren times open paren 3 x plus 2 close paren plus 5

12x2+2x−9=(4x−2)(3x−1)−11
12 x squared plus 2 x minus 9 is equal to open paren 4 x minus 2 close paren times open paren 3 x minus 1 close paren minus 11

12x2+2x−9=(4x−2)(3x+2)−5
12 x squared plus 2 x minus 9 is equal to open paren 4 x minus 2 close paren times open paren 3 x plus 2 close paren minus 5

12x2+2x−9=(4x−2)(3x+2)−13

Answer 1

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

1. **Divide the leading term**:
- Divide \( 12x^2 \) by \( 4x \) which gives \( 3x \).

2. **Multiply**:
- Multiply \( 3x \) by \( 4x - 2 \) to get \( 12x^2 - 6x \).

3. **Subtract**:
- Subtract \( (12x^2 - 6x) \) from \( 12x^2 + 2x - 9 \):
\[
(12x^2 + 2x - 9) - (12x^2 - 6x) = 8x - 9
\]

4. **Repeat**:
- Divide the leading term \( 8x \) by \( 4x \) which gives \( 2 \).
- Multiply \( 2 \) by \( 4x - 2 \) to get \( 8x - 4 \).
- Subtract \( (8x - 4) \) from \( 8x - 9 \):
\[
(8x - 9) - (8x - 4) = -5
\]

So the quotient is \( 3x + 2 \) and the remainder is \(-5\).

Thus, we can express the original polynomial as:

\[
12x^2 + 2x - 9 = (4x - 2)(3x + 2) - 5
\]

So the correct response is:

**12x² + 2x - 9 = (4x - 2)(3x + 2) - 5**