To divide the polynomial \( 12x^2 + 2x - 9 \) by \( 4x - 2 \), we can use polynomial long division.
Step 1: Divide the first term of the dividend by the first term of the divisor.
The first term of the dividend \( 12x^2 \) divided by the first term of the divisor \( 4x \) gives:
\[ \frac{12x^2}{4x} = 3x \]
Step 2: Multiply the entire divisor by \( 3x \) and subtract from the dividend.
Now we multiply \( 3x \) by the divisor \( 4x - 2 \):
\[ 3x(4x - 2) = 12x^2 - 6x \]
Next, we subtract this from the original polynomial:
\[ (12x^2 + 2x - 9) - (12x^2 - 6x) = 2x + 6x - 9 = 8x - 9 \]
Step 3: Repeat the process with the new polynomial \( 8x - 9 \).
Now divide the first term \( 8x \) by \( 4x \):
\[ \frac{8x}{4x} = 2 \]
Now we multiply the entire divisor \( 4x - 2 \) by \( 2 \):
\[ 2(4x - 2) = 8x - 4 \]
Next, we subtract this from the polynomial we obtained, \( 8x - 9 \):
\[ (8x - 9) - (8x - 4) = -9 + 4 = -5 \]
At this point, we cannot divide further since the remainder \(-5\) has a lower degree than the divisor \(4x - 2\).
Conclusion:
Now we can express the result. The quotient is \(3x + 2\) and the remainder is \(-5\).
Thus, we have:
\[ 12x^2 + 2x - 9 = (4x - 2)(3x + 2) - 5 \]
So the correct response is:
12x² + 2x − 9 = (4x − 2)(3x + 2) − 5