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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 minus 5

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
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
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1 answer

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

  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 product from the original polynomial: \[ (12x^2 + 2x - 9) - (12x^2 - 6x) = 8x - 9 \]

  4. Repeat the process with the new polynomial \( 8x - 9 \):

    • Divide the leading term: \[ \frac{8x}{4x} = 2 \]

    • Multiply the divisor: \[ 2(4x - 2) = 8x - 4 \]

    • Subtract again: \[ (8x - 9) - (8x - 4) = -5 \]

Now we have completed the division, resulting in a quotient of \( 3x + 2 \) and a remainder of \( -5 \).

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

The correct answer is: \[ 12x^2 + 2x - 9 = (4x - 2)(3x + 2) - 5 \]