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.

Question

GPT-4o mini · Answer

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 $-5$.