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 \(420x - 13\).