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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)−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

1 answer

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