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)−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+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+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−1)−11

Answer 1

To find the quotient and remainder when dividing the polynomial $12x^2+2x-9$ by $4x-2$, we can use polynomial long division.

Steps to Divide:

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

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

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

4. Now repeat the process with the new polynomial $8x-9$: $$\frac{8x}{4x}=2$$

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

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

Final Results:

The quotient is $3x+2$ and the remainder is $-5$.

Thus, we can express the polynomial as: $$12x^2+2x-9=(4x-2)(3x+2)-5$$

Confirming the Response:

The correct expression based on the calculations is: $$12x^2+2x-9=(4x-2)(3x+2)-5$$

Therefore, the correct answer is: 12x² + 2x - 9 = (4x - 2)(3x + 2) - 5.

Answer 2

The image shows the first few steps of the polynomial division (3x4+9x2−13)÷(x2−5x). Complete the division process and find the remainder of this polynomial division. (1 point) Responses The remainder is 407x. The remainder is 407 x . The remainder is 420x−13. The remainder is 420 x minus 13 . The remainder is 407. The remainder is 407. The remainder is 2,087.

Answer 3

To complete the division of the polynomial $3x^4+9x^2-13$ by $x^2-5x$, we will perform polynomial long division step by step.

Step-by-Step Polynomial Division:

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

2. Multiply the entire divisor by $3x^2$: $$3x^2(x^2-5x)=3x^4-15x^3$$

3. Subtract from the original polynomial: $$(3x^4+9x^2-13)-(3x^4-15x^3)=15x^3+9x^2-13$$

4. Repeat with the new polynomial, $15x^3+9x^2-13$:

   • Divide the leading term: $$\frac{15x^3}{x^2}=15x$$

   • Multiply the divisor: $$15x(x^2-5x)=15x^3-75x^2$$

   • Subtract: $$(15x^3+9x^2-13)-(15x^3-75x^2)=9x^2+75x^2-13=84x^2-13$$

5. Repeat with the new polynomial, $84x^2-13$:

   • Divide the leading term: $$\frac{84x^2}{x^2}=84$$

   • Multiply the divisor: $$84(x^2-5x)=84x^2-420x$$

   • Subtract: $$(84x^2-13)-(84x^2-420x)=420x-13$$

So at this point, we have divided completely, and the next divisor $420x-13$ has a degree smaller than $x^2-5x$.

Conclusion:

Thus, the remainder of the polynomial division is: $$\text{The remainder is }420x-13.$$

Therefore, the correct response is: The remainder is $420x-13$.