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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 plus 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+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

1 answer

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

Steps for Long Division:

  1. Divide the Leading Terms: \[ \frac{12x^2}{4x} = 3x \]

  2. Multiply by the Divisor: \[ 3x(4x - 2) = 12x^2 - 6x \]

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

  4. Repeat the Process:

    • Divide the leading terms: \[ \frac{8x}{4x} = 2 \]
    • Multiply by the divisor: \[ 2(4x - 2) = 8x - 4 \]
    • Subtract: \[ (8x - 9) - (8x - 4) = -5 \]

Now we have a quotient of \(3x + 2\) and a remainder of \(-5\).

Final Expression:

The polynomial can be expressed as: \[ 12x^2 + 2x - 9 = (4x - 2)(3x + 2) - 5 \]

So, the correct response is: \[ 12x^2 + 2x - 9 = (4x - 2)(3x + 2) - 5 \]