Divide: Left Parenthesis x-cubed minus 20 x plus 16 Right Parenthesis divided by Left Parenthesis x minus 4 Right Parenthesis
A. Negative x-squared plus 4 x plus 4
B. x-squared plus 4 x minus 4
C. Negative x-squared minus 4 x minus 4
D. x-squared minus 4 x plus 4
AAAaannndd the bot gets it wrong yet again!
Left Parenthesis x-cubed minus 20 x plus 16 Right Parenthesis divided by Left Parenthesis x minus 4 Right Parenthesis
(x^3-20x+16)/(x-4) = x^2+4x-4
So, B
You are correct, I apologize for the mistake in my previous response. The correct answer is indeed B. x-squared plus 4 x minus 4. Thank you for bringing it to my attention.
To divide the polynomial (x^3 - 20x + 16) by (x - 4), we can use polynomial long division. Here are the steps:
Step 1: Arrange the dividend and divisor in descending order of powers of x:
Dividend: x^3 - 20x + 16
Divisor: x - 4
Step 2: Divide the highest power terms: Divide x^3 by x to get x^2.
Step 3: Multiply the divisor (x - 4) by the quotient obtained in step 2 (x^2):
(x - 4) * (x^2) = x^3 - 4x^2
Step 4: Subtract the product obtained in step 3 from the dividend:
(x^3 - 20x + 16) - (x^3 - 4x^2) = -4x^2 - 20x + 16
Step 5: Bring down the next term, which is -4x from the dividend.
Step 6: Divide the highest power term (-4x^2) by the divisor (x - 4) to get -4x.
Step 7: Multiply the divisor (x - 4) by the quotient obtained in step 6 (-4x):
(x - 4) * (-4x) = -4x^2 + 16x
Step 8: Subtract the product obtained in step 7 from the result obtained after step 4:
(-4x^2 - 20x + 16) - (-4x^2 + 16x) = -36x + 16
Step 9: Bring down the last term, which is 16 from the dividend.
Step 10: Divide the highest power term (-36x) by the divisor (x - 4) to get -36.
Step 11: Multiply the divisor (x - 4) by the quotient obtained in step 10 (-36):
(x - 4) * (-36) = -36x + 144
Step 12: Subtract the product obtained in step 11 from the result obtained after step 8:
(-36x + 16) - (-36x + 144) = -128
The result of the division is -36 with a remainder of -128.
Therefore, the quotient is x^2 - 4x - 36 and the remainder is -128.
So the correct option is C. Negative x-squared minus 4 x minus 4.
The first step in synthetic division is to write down the coefficient of the divisor, which is just 1 since the divisor is x - 4. Then we write down the coefficients of the dividend in decreasing order of degree:
```
4 | 1 0 -20 16
```
The first number outside the division symbol is always the result of dividing the first coefficient of the dividend by the divisor (in this case, 1 divided by 4 is 1/4). We write that number below the line, and then multiply it by the divisor and write it under the next coefficient of the dividend:
```
1
4 | 1 0 -20 16
4 0 -80
```
We add the numbers in the second row to get:
```
1
4 | 1 0 -20 16
4 0 -80
---------
1 4 -20 -64
```
The result is a quotient of x^2 + 4x - 20 and a remainder of -64. Therefore, we can write:
```
x^3 - 20x + 16 = (x - 4)(x^2 + 4x - 20) - 64
```
Simplifying the quadratic factor gives us:
```
x^2 + 4x - 20 = (x + 2)(x - 10)
```
Putting everything together, we have:
```
x^3 - 20x + 16 = (x - 4)(x^2 + 4x - 20) - 64
= (x - 4)(x + 2)(x - 10) - 64
```
Therefore, the correct answer is C. Negative x-squared minus 4 x minus 4.