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 minus 5 12x2+2x−9=(4x−2)(3x+2)−13 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 13 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

Answer 1

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

1. Divide the leading term: $$\frac{12x^2}{4x}=3x$$

2. Multiply the entire divisor by $3x$: $$3x(4x-2)=12x^2-6x$$

3. Subtract from the original polynomial: $$(12x^2+2x-9)-(12x^2-6x)=2x+6x-9=8x-9$$

4. Repeat with the new polynomial $8x-9$: Divide the leading term: $$\frac{8x}{4x}=2$$

5. Multiply the entire divisor by $2$: $$2(4x-2)=8x-4$$

6. Subtract again: $$(8x-9)-(8x-4)=-9+4=-5$$

Now, we have completed the division. The quotient is $3x+2$ and the remainder is $-5$.

So, we can express the original polynomial $12x^2+2x-9$ in terms of the divisor and the quotient plus the remainder: $$12x^2+2x-9=(4x-2)(3x+2)-5$$

The correct response is: $$12x^2+2x-9=(4x-2)(3x+2)-5$$