Divide. (2x3 – x2 – 24x + 12) ÷ (2x – 1)
A. x2 – 1
B. x2 – 12
C. 2x – 1
D. x squared minus one half x
Don't understand why the bot cannot do this question..
(2x3 – x2 – 24x + 12) ÷ (2x – 1) = x^2 - 12
To divide the polynomial (2x^3 - x^2 - 24x + 12) by the polynomial (2x - 1), we can use polynomial long division. Here's how to do it step by step:
Step 1: Arrange the terms in descending order of their degrees.
The dividend polynomial: 2x^3 - x^2 - 24x + 12
The divisor polynomial: 2x - 1
Step 2: Divide the first term of the dividend polynomial (2x^3) by the first term of the divisor polynomial (2x). The result is x^2.
Write x^2 above the division bar.
x^2
_____________
2x - 1 | 2x^3 - x^2 - 24x + 12
Step 3: Multiply the divisor polynomial (2x - 1) by the quotient obtained in Step 2 (x^2).
The result of the multiplication is 2x^3 - x^2.
x^2
_____________
2x - 1 | 2x^3 - x^2 - 24x + 12
- (2x^3 - x^2)
_________________
- 24x
Step 4: Subtract the result of Step 3 from the dividend polynomial.
The subtraction is -24x - (-24x) = 0.
x^2
_____________
2x - 1 | 2x^3 - x^2 - 24x + 12
- (2x^3 - x^2)
_________________
- 24x
- (-24x)
_________________
0
Step 5: Bring down the next term from the dividend polynomial (12).
Write 12 next to the -24x.
x^2
_____________
2x - 1 | 2x^3 - x^2 - 24x + 12
- (2x^3 - x^2)
_________________
- 24x
- (-24x)
_________________
0
+ 12
Step 6: Divide the resulting polynomial (-24x + 12) by the divisor polynomial (2x - 1).
The result is -12.
x^2 - 12
_____________
2x - 1 | 2x^3 - x^2 - 24x + 12
- (2x^3 - x^2)
_________________
- 24x
- (-24x)
_________________
0
+ 12
- (12x - 12)
______________________
0
Step 7: Since the remainder is zero, we have completed the division.
Therefore, the quotient of (2x^3 - x^2 - 24x + 12) divided by (2x - 1) is x^2 - 12.
The correct answer is B. x^2 - 12.
To divide (2x^3 – x^2 – 24x + 12) by (2x – 1), you can use polynomial long division.
First, make sure the polynomial is arranged in descending order:
2x^3 – x^2 – 24x + 12
Now, let's start the polynomial long division:
x^2 + 6x + 6
-------------------------------
2x - 1 | 2x^3 - x^2 - 24x + 12
To start, divide the first terms of the polynomials: (2x^3)/(2x) = x^2
Then, multiply the divisor (2x - 1) by the result you just obtained (x^2):
x^2
-------------------------------
2x - 1 | 2x^3 - x^2 - 24x + 12
- (2x^3 - x^2)
This cancels out the first term of the dividend. Now subtract the multiplied result from the original dividend:
x^2
-------------------------------
2x - 1 | 2x^3 - x^2 - 24x + 12
- (2x^3 - x^2)
-------------------------------
-25x + 12
Next, bring down the next term from the dividend (-25x), and the process continues:
x^2 + 6x
-------------------------------
2x - 1 | 2x^3 - x^2 - 24x + 12
- (2x^3 - x^2)
-------------------------------
-25x + 12
- (-25x + 12)
-------------------------------
0
Since the remainder is zero, the result of the division is x^2 + 6x + 6.
Therefore, the correct answer is:
A. x^2 – 1