what is the remainder when the function
f(x)=x^4-3x^3+7x-1 is divided by (x-2)
A. 2
B. 5
C. 17
D. 25
Use the factor theorem to determine if (x-4) is a factor of x^5-3x^4-x-3
A. (x-4) is a factor because f(4)=0
B. (x-4) is a factor because f(4)≠0
C. (x-4) is not a factor because f(4)=0
D. (x-4) is not a factor because f(4)≠0
To find the remainder when f(x) = x^4 - 3x^3 + 7x - 1 is divided by (x-2), we can use synthetic division.
We set up the division with 2 as the divisor:
2 | 1 -3 7 -1
-----------------------
2 -2 10 19
The remainder is the last value in the bottom row, which is 19.
Therefore, the remainder when f(x) is divided by (x-2) is 19.
To determine if (x-4) is a factor of f(x) = x^5 - 3x^4 - x - 3, we can use the factor theorem.
The factor theorem states that if (x-a) is a factor of f(x), then f(a) = 0.
To use this theorem, we substitute the value of a = 4 into f(x) and check if f(4) = 0.
f(4) = (4)^5 - 3(4)^4 - (4) - 3
= 1024 - 3(256) - 4 - 3
= 1024 - 768 - 7
= 249
Since f(4) is not equal to 0, (x-4) is not a factor of f(x).
Therefore, the answer is D. (x-4) is not a factor because f(4)≠0.
To find the remainder when a function is divided by a binomial, you can apply the Remainder Theorem.
For the first question, to find the remainder when f(x) = x^4 - 3x^3 + 7x - 1 is divided by (x - 2), you substitute the value of x in the divisor, which is 2, into the function f(x) and evaluate it.
f(2) = (2)^4 - 3(2)^3 + 7(2) - 1 = 16 - 24 + 14 - 1 = 5
Therefore, the remainder is 5. So, the correct answer is B. 5.
For the second question, you can use the Factor Theorem to determine if (x - 4) is a factor of x^5 - 3x^4 - x - 3.
The Factor Theorem states that if a polynomial function f(x) has a factor (x - k), then f(k) = 0.
To check if (x - 4) is a factor, substitute the value of k, which is 4, into the function f(x) and evaluate it.
f(4) = (4)^5 - 3(4)^4 - 4 - 3 = 1024 - 192 - 4 - 3 = 825
Since f(4) ≠ 0, we conclude that the function f(x) does not have (x - 4) as a factor. Therefore, the correct answer is D. (x - 4) is not a factor because f(4) ≠ 0.
To find the remainder when the function f(x) = x^4 - 3x^3 + 7x - 1 is divided by (x - 2), you can use the synthetic division method.
Step 1: Set up the synthetic division table, with 2 on the left side as the divisor:
2 | 1 -3 7 -1
Step 2: Bring down the first coefficient, which is 1, to the second row:
2 | 1 -3 7 -1
|
| 1
Step 3: Multiply 2 (the divisor) by 1 (the value in the second row) and place the result in the third row:
2 | 1 -3 7 -1
| 2
| 1 2
Step 4: Add -3 (the value in the second row) and 2 (the value in the third row), and place the result in the fourth row:
2 | 1 -3 7 -1
| 2
| 1 2
| -1
Step 5: Multiply 2 (the divisor) by -1 (the value in the fourth row) and place the result in the fifth row:
2 | 1 -3 7 -1
| 2
| 1 2
| -1
| -2
Step 6: Add 7 (the value in the third row) and -2 (the value in the fifth row), and place the result in the sixth row:
2 | 1 -3 7 -1
| 2
| 1 2
| -1
| -2
| 5
The remainder is 5.
Therefore, the answer is B. 5.
To determine if (x - 4) is a factor of f(x) = x^5 - 3x^4 - x - 3, we can use the factor theorem.
If we substitute x = 4 into the function and get the value as zero, then (x - 4) is a factor.
So, let's evaluate f(4):
f(4) = (4)^5 - 3(4)^4 - (4) - 3
= 1024 - 3(256) - 4 - 3
= 1024 - 768 - 7
= 249
Since f(4) ≠ 0, (x - 4) is not a factor.
Therefore, the answer is D. (x - 4) is not a factor because f(4) ≠ 0.