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Question

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 plus 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 minus 5

12x2+2x−9=(4x−2)(3x+2)−13

Answer 1

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

1. Divide the leading term:

   • Divide \( 12x^2 \) by \( 4x \) which gives \( 3x \).

2. Multiply:

   • Multiply \( 3x \) by \( 4x - 2 \) to get \( 12x^2 - 6x \).

3. Subtract:

   • Subtract \( (12x^2 - 6x) \) from \( 12x^2 + 2x - 9 \): \[ (12x^2 + 2x - 9) - (12x^2 - 6x) = 8x - 9 \]

4. Repeat:

   • Divide the leading term \( 8x \) by \( 4x \) which gives \( 2 \).
   • Multiply \( 2 \) by \( 4x - 2 \) to get \( 8x - 4 \).
   • Subtract \( (8x - 4) \) from \( 8x - 9 \): \[ (8x - 9) - (8x - 4) = -5 \]

So the quotient is \( 3x + 2 \) and the remainder is \(-5\).

Thus, we can express the original polynomial as:

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

So the correct response is:

12x² + 2x - 9 = (4x - 2)(3x + 2) - 5