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 $-5$.