Which binomial is a factor of f(x) = x4 - 27x2 -14x + 120?
(x-4)
(x-2)
(x + 5)
(x + 6)
From all the other answers I have given you ...
What do you think you should try ?
Factor x3 4x2 3x + 18 = 0, given that 3 is a zero.
Option A: (x + 2)(x 3)2 = 0
Option B: (x 2)(x 3)2 = 0
Option C: (x 2)(x 3)(x + 3) = 0
Option D:
To determine if a binomial is a factor of the polynomial f(x), we can use the Remainder Theorem. According to the Remainder Theorem, if a binomial (x - c) is a factor of f(x), then f(c) will be equal to zero.
Let's test the given binomials one by one:
1. (x - 4):
To check if (x - 4) is a factor, we substitute x = 4 into f(x):
f(4) = (4)^4 - 27(4)^2 - 14(4) + 120
= 256 - 27(16) - 56 + 120
= 256 - 432 - 56 + 120
= -112
Since f(4) is not equal to zero, (x - 4) is not a factor.
2. (x - 2):
To check if (x - 2) is a factor, we substitute x = 2 into f(x):
f(2) = (2)^4 - 27(2)^2 - 14(2) + 120
= 16 - 27(4) - 28 + 120
= 16 - 108 - 28 + 120
= 0
Since f(2) is equal to zero, (x - 2) is a factor.
3. (x + 5):
To check if (x + 5) is a factor, we substitute x = -5 into f(x):
f(-5) = (-5)^4 - 27(-5)^2 - 14(-5) + 120
= 625 - 27(25) + 70 + 120
= 625 - 675 + 70 + 120
= 140
Since f(-5) is not equal to zero, (x + 5) is not a factor.
4. (x + 6):
To check if (x + 6) is a factor, we substitute x = -6 into f(x):
f(-6) = (-6)^4 - 27(-6)^2 - 14(-6) + 120
= 1296 - 27(36) + 84 + 120
= 1296 - 972 + 84 + 120
= 528
Since f(-6) is not equal to zero, (x + 6) is not a factor.
Therefore, the only binomial that is a factor of f(x) = x^4 - 27x^2 - 14x + 120 is (x - 2).
To determine which binomial is a factor of the given polynomial, you can use synthetic division. In synthetic division, you divide the polynomial by the binomial, and if the remainder is zero, then the binomial is a factor.
Let's start with the binomial (x - 4).
1. Set up the synthetic division table by writing down the coefficients of the polynomial: | 1 | -27 | -14 | 120 |
2. Write down the root of the binomial (x - 4) as a positive number and place it outside the division bar: 4 | 1 | -27 | -14 | 120 |
3. Bring down the first coefficient (1) and multiply it by the root (4), then write the result below the second coefficient (-27): 4 | 1 | -27 | -14 | 120 |
4 | 4 |
4. Add the result of the previous step with the second coefficient (-27): 4 | 1 | -27 | -14 | 120 |
4 | 4 |
--------------
1 | -23 |
5. Bring down the next coefficient (-14), and repeat the process: 4 | 1 | -27 | -14 | 120 |
4 | 4 |
--------------
1 | -23 | -38 |
-56 |
6. Finally, bring down the last coefficient (120), add it to the result, and multiply it by the root: 4 | 1 | -27 | -14 | 120 |
4 | 4 |
--------------
1 | -23 | -38 | 82 |
------
1 | -23 | -38 | 82 |
The remainder is 82, so (x - 4) is not a factor of f(x) = x^4 - 27x^2 - 14x + 120.
Repeat the process with the other binomials (x - 2), (x + 5), and (x + 6) to find the binomial that is a factor of f(x).
This step-by-step process will help you determine which binomial is a factor of the given polynomial.