If f, left bracket, x, right bracket, equals, 4, x, to the power 4 , plus, x, cubed, minus, 4f(x)=4x
4
+x
3
−4, then what is the remainder when f, left bracket, x, right bracketf(x) is divided by x, minus, 1x−1?
To find the remainder when f(x) is divided by x - 1, we can use the remainder theorem. According to the remainder theorem, if f(a) is divided by x - a, then the remainder is equal to f(a).
In this case, we want to find the remainder when f(x) is divided by x - 1. So, we need to find f(1).
f(x) = 4x^4 + x^3 - 4
Substituting x = 1 into f(x), we have:
f(1) = 4(1)^4 + (1)^3 - 4
= 4 + 1 - 4
= 1
Therefore, the remainder when f(x) is divided by x - 1 is 1.
To find the remainder when f(x) is divided by x-1, we can use synthetic division.
1. Set up the synthetic division table using the coefficients of f(x) in descending order:
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| | 4 | 1 | -4 | 4 |
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| -1 | | | | |
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2. Bring down the 4:
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| | 4 | 1 | -4 | 4 |
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| -1 | | | | 4 |
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3. Multiply -1 by 4, and write the result below 1:
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| | 4 | 1 | -4 | 4 |
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| -1 | | -4 | | 4 |
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4. Add 1 and -4, and write the result below -4:
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| | 4 | 1 | -4 | 4 |
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| -1 | | -4 | -3 | 4 |
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5. Multiply -1 by -3, and write the result below -4:
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| | 4 | 1 | -4 | 4 |
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| -1 | | -4 | -3 | 4 |
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| 3 | 3 |
6. Add -4 and 3, and write the result below -3:
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| | 4 | 1 | -4 | 4 |
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| -1 | | -4 | -3 | 4 |
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| 3 | 3 |
| | 1 |
7. The remainder is the number at the bottom right of the synthetic division table. In this case, the remainder is 1.
Therefore, the remainder when f(x) is divided by x-1 is 1.
To find the remainder when dividing f(x) by x-1, we can use the remainder theorem. The remainder theorem states that if we substitute the divisor (x-1) into the function f(x) and evaluate it, the result will be the remainder.
In this case, we need to find f(x) / (x-1), and the goal is to convert the function f(x) into the form of (x - 1) * q(x) + r, where q(x) is the quotient and r is the remainder.
We are given f(x) = 4x^4 + x^3 - 4x + 4. To find the remainder, we divide f(x) by (x-1).
We can use polynomial long division or synthetic division to perform the division. Let's use synthetic division for simplicity.
The divisor is (x-1), which means we substitute x = 1 into the function.
Step 1: Set up the division:
_____________________________________
(x - 1) | 4x^4 + x^3 - 4x + 4
Step 2: Bring down the first term, which is 4x^4.
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(x - 1) | 4x^4 + x^3 - 4x + 4
4x^3
Step 3: Multiply the divisor (x - 1) by the quotient obtained (4x^3), and subtract it from the next term of the dividend.
_____________________________________
(x - 1) | 4x^4 + x^3 - 4x + 4
(-4x^4)
4x^3
______________________________
0 + x^3 - 4x + 4
Step 4: Bring down the next term, which is x^3.
_____________________________________
(x - 1) | 4x^4 + x^3 - 4x + 4
(-4x^4)
4x^3
______________________________
0 + x^3 - 4x + 4
x^3
Step 5: Multiply the divisor (x - 1) by the quotient obtained (x^3), and subtract it from the next term of the dividend.
_____________________________________
(x - 1) | 4x^4 + x^3 - 4x + 4
(-4x^4)
4x^3
______________________________
0 + x^3 - 4x + 4
x^3
(- x^3)
______________________________
0 - 4x + 4
Step 6: Bring down the next term, which is -4x.
_____________________________________
(x - 1) | 4x^4 + x^3 - 4x + 4
(-4x^4)
4x^3
______________________________
0 + x^3 - 4x + 4
x^3
(- x^3)
______________________________
0 - 4x + 4
(4x)
______________________________
0 - 0 + 4
Step 7: There are no more terms to bring down. The remainder is 4.
So, when we divide f(x) by (x-1), the remainder is 4.