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.
To divide the polynomial 12x^2 + 2x - 9 by 4x - 2, we can use long division:
3x + 2
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4x - 2 | 12x^2 + 2x - 9
- (12x^2 - 6x)
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8x - 9
- (8x - 4)
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-5
The quotient is 3x + 2 and the remainder is -5. Thus, the polynomial can be expressed as the product of the divisor and the quotient plus the remainder as:
12x^2 + 2x - 9 = (4x - 2)(3x + 2) - 5
To divide the polynomial 12x^2 + 2x - 9 by 4x - 2, we can use the polynomial long division method.
Step 1: Divide the highest degree term of the dividend by the highest degree term of the divisor.
12x^2 / 4x = 3x
Step 2: Multiply the quotient obtained in step 1 by the divisor.
3x * (4x - 2) = 12x^2 - 6x
Step 3: Subtract the product obtained in step 2 from the dividend.
(12x^2 + 2x - 9) - (12x^2 - 6x) = 8x - 9
Step 4: Repeat the process with the new dividend (8x - 9).
Step 5: Divide the highest degree term of the new dividend by the highest degree term of the divisor.
8x / 4x = 2
Step 6: Multiply the quotient obtained in step 5 by the divisor.
2 * (4x - 2) = 8x - 4
Step 7: Subtract the product obtained in step 6 from the new dividend.
(8x - 9) - (8x - 4) = -5
Since the degree of the new dividend (-5) is less than the degree of the divisor, we have reached the end of the division process.
Therefore, the quotient is 3x + 2 and the remainder is -5.
Expressing the polynomial as the product of the divisor and the quotient plus the remainder:
12x^2 + 2x - 9 = (4x - 2)(3x + 2) - 5