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.
To divide the polynomial 12x^2 + 2x - 9 by 4x - 2, we use long division.
3x + 2
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4x - 2 | 12x^2 + 2x - 9
- (12x^2 - 6x)
-----------------
8x - 9
Therefore, the quotient is 3x + 2 and the remainder is 8x - 9.
So, the polynomial can be expressed as the product of the divisor and the quotient plus the remainder as:
(4x - 2)(3x + 2) + (8x - 9)
To divide the polynomial 12x^2+2x−9 by 4x−2, follow these steps:
Step 1: Write the dividend (the polynomial being divided) and the divisor (the polynomial you're dividing by) in long division format:
_______________
4x−2 | 12x^2 + 2x - 9
Step 2: Divide the first term of the dividend by the first term of the divisor:
12x^2 ÷ 4x = 3x
Step 3: Multiply the divisor by the result from Step 2:
3x * (4x−2) = 12x^2 - 6x
Step 4: Subtract the result from Step 3 from the original dividend:
12x^2 + 2x - 9 - (12x^2 - 6x) = 2x - 9 + 6x = 8x - 9
Step 5: Bring down the next term from the original dividend:
_______________
4x−2 | 12x^2 + 2x - 9
8x - 9
Step 6: Repeat steps 2-4 with the new numerator 8x - 9:
(8x-9) / (4x-2) = 2
Step 7: Multiply the divisor by the result from Step 6:
2 * (4x-2) = 8x - 4
Step 8: Subtract the result from Step 7 from the new numerator:
8x - 9 - (8x - 4) = 8x - 9 - 8x + 4 = -5
Since the new numerator (-5) is of lesser degree than the divisor (4x−2), we stop here.
Finally, the polynomial 12x^2+2x−9 divided by 4x−2 can be expressed as:
12x^2 + 2x - 9 = (4x - 2)(3x + 2) + (-5)
Therefore, the polynomial is the product of the divisor and the quotient, with the remainder (-5).