Divide the polynomial 12x^2+2x−9 by 4x−2. Then express the polynomial as the product of the divisor and the quotient plus the remainder. (1 point)
12x^2+2x−9=(4x−2)(3x+2)−5
12x^2+2x−9=(4x−2)(3x+2)+5
12x^2+2x−9=(4x−2)(3x−1)−11
12x^2+2x−9=(4x−2)(3x+2)−13
12x^2+2x−9=(4x−2)(3x+2)+5
To divide the polynomial 12x^2+2x−9 by 4x−2, you can use the synthetic division method. Here are the steps to follow:
Step 1: Arrange the polynomial in descending order of exponents.
The polynomial is already in this form: 12x^2+2x−9.
Step 2: Set up the synthetic division table.
Place the divisor, 4x−2, as the divisor on the left side and write the coefficients of the polynomial as the dividend on the right side. Leave an extra space for the remainder.
| 4x−2 |
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| 12 2 −9
Step 3: Perform synthetic division.
Bring down the first coefficient, 12, and multiply it by the divisor, 4x−2. Write the product below the second coefficient.
| 4x−2 |
-------------------------------
| 12 2 −9
48x - 24
Add the resulting terms vertically to get the new polynomial.
| 4x−2 |
-------------------------------
| 12 2 −9
48x - 24
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| 12 50x - 33
Step 4: Write the final result.
The divided polynomial is given by (4x−2)(12 + 50x−33 / (4x−2)). This can be further simplified as (4x−2)(3x+2)−(33 / 4x−2).
So, the polynomial 12x^2+2x−9 can be expressed as the product of the divisor 4x−2 and the quotient 3x+2, plus the remainder −33 / (4x−2).