Divide
(2x3 – x2 – 24x + 12) ÷ (2x – 1)
2 x^3 – x^2 – 24 x + 12)
= (2 x^3 -24 x) - (x^2 - 12)
= 2 x (x^2-12) - 1 (x^2-12)
= (2x-1)(x^2-12)
now divide that by (2x-1) :)
Thank you, I always appreciate knowing the steps that leads to the solution that I can work.
You are welcome. You could have done this by long division but often you can see the factors more quickly.
The answer is x^2−12 BTW
To divide the polynomial (2x^3 – x^2 – 24x + 12) by the binomial (2x – 1), you can use polynomial long division. Here's how:
Step 1: Arrange the dividend and divisor in descending order, with missing terms represented by zeros. In this case, the dividend is 2x^3 – x^2 – 24x + 12 and the divisor is 2x – 1. Rewrite the dividend as: 2x^3 + 0x^2 – x^2 – 24x + 12.
Step 2: Divide the first term of the dividend (2x^3) by the first term of the divisor (2x) to get the quotient term. In this case, the quotient term is x^2.
Step 3: Multiply the divisor (2x – 1) by the quotient term (x^2) and write the result under the dividend, aligning like terms. You will get 2x^3 – x^2.
Step 4: Subtract the result obtained in step 3 from the dividend. So, subtract 2x^3 – x^2 from 2x^3 + 0x^2 – x^2 – 24x + 12. You are left with 0x^3 + x^2 – 24x + 12.
Step 5: Bring down the next term from the dividend (in this case, it is -24x). You now have x^2 – 24x + 12.
Step 6: Repeat steps 2-4 with the new polynomial (x^2 – 24x + 12) and the divisor (2x – 1). Divide the first term of the new polynomial (x^2) by the first term of the divisor (2x) to get the next quotient term. You should obtain the quotient term 1/2.
Step 7: Multiply the divisor (2x – 1) by the new quotient term (1/2) and write the result under the new polynomial, aligning like terms. You will get x^2 – x.
Step 8: Subtract the result obtained in step 7 from the new polynomial. So, subtract x^2 – x from x^2 – 24x + 12. You are left with -x - 24x + 12.
Step 9: Bring down the next term from the new polynomial (-x - 24x + 12). You now have -25x + 12.
Step 10: Repeat steps 2-4 with the new polynomial (-25x + 12) and the divisor (2x – 1). Divide the first term of the new polynomial (-25x) by the first term of the divisor (2x) to get the next quotient term. You should obtain the quotient term -12.5.
Step 11: Multiply the divisor (2x – 1) by the new quotient term (-12.5) and write the result under the new polynomial, aligning like terms. You will get -25x + 12.5.
Step 12: Subtract the result obtained in step 11 from the new polynomial. So, subtract -25x + 12.5 from -25x + 12. You are left with -0.5.
Since there are no more terms to bring down, you have completed the division. The quotient is x^2 + 0.5x - 12.5, and the remainder is -0.5.
Therefore, (2x^3 – x^2 – 24x + 12) ÷ (2x – 1) equals x^2 + 0.5x - 12.5 with a remainder of -0.5.