(2x^3 - x^2 - 24x + 12) divided by (2x - 1)
(2x^3 - x^2 - 24x + 12) divided by (2x - 1)
= ( x^2(2x-1) - 12(2x-1) )/(2x-1)
= (2x-1)(x^2 - 12)/(2x-1)
= x^2 - 12 , x ≠ 1/2
To divide the polynomial (2x^3 - x^2 - 24x + 12) by (2x - 1), we can use long division. Here are the step-by-step instructions:
1. Arrange the terms of the dividend (2x^3 - x^2 - 24x + 12) in descending order of the exponent:
2x^3 - x^2 - 24x + 12
2. Divide the first term of the dividend by the first term of the divisor:
2x^3 / (2x) = x^2
3. Multiply the divisor (2x - 1) by the quotient we just found (x^2):
(2x - 1) * (x^2) = 2x^3 - x^2
4. Subtract the product obtained in step 3 from the dividend:
(2x^3 - x^2 - 24x + 12) - (2x^3 - x^2) = - 24x + 12
5. Bring down the next term from the dividend (in this case, -24x):
- 24x
6. Divide the first term of the new dividend (-24x) by the first term of the divisor (2x):
-24x / (2x) = -12
7. Multiply the divisor (2x - 1) by the new quotient we found (-12):
(2x - 1) * (-12) = -24x + 12
8. Subtract the product obtained in step 7 from the new dividend (-24x):
(-24x) - (-24x + 12) = 0
9. Since the subtraction results in zero, there is no remainder.
10. Write the quotient obtained in steps 2 and 6:
Quotient = x^2 - 12
Therefore, the result of dividing (2x^3 - x^2 - 24x + 12) by (2x - 1) is x^2 - 12.