Use the remainder theorem to find the remainder when f(x) is divided by x -2. Then use the factor theorem to determine whether x -2 is a factor of f(x)
f(x) = 2x^5 + 6x^4 - x^2 - 7x - 12
Thank you!
f(2) = 2(32) + 6(16) - 4 - 14 - 12 = 130
so c-2 is not a factor
by synthetic division, (hard to show in this format)
2 | 2 6 0 -1 -7 -12
....-------------
........4 20 40 78 142
....210 20 39 71 130
I guess I should have done the division first
To find the remainder when f(x) is divided by x - 2, we can use the remainder theorem.
According to the remainder theorem, if we substitute the value x = 2 into f(x), the resulting value will be the remainder when f(x) is divided by x - 2.
Let's substitute x = 2 into f(x):
f(2) = 2(2)^5 + 6(2)^4 - (2)^2 - 7(2) - 12
= 2(32) + 6(16) - 4 - 14 - 12
= 64 + 96 - 4 - 14 - 12
= 144 - 30
= 114
Therefore, the remainder when f(x) is divided by x - 2 is 114.
Now, let's use the factor theorem to determine whether x - 2 is a factor of f(x).
According to the factor theorem, if the remainder is zero when f(x) is divided by x - 2, then x - 2 is a factor of f(x). Conversely, if the remainder is non-zero, then x - 2 is not a factor of f(x).
Since the remainder we found earlier is 114 (non-zero), it means that x - 2 is not a factor of f(x).
To find the remainder when f(x) is divided by x - 2, we can use the remainder theorem. According to the remainder theorem, if a polynomial f(x) is divided by x - a, then the remainder is equal to f(a).
In this case, we need to find the remainder when f(x) = 2x^5 + 6x^4 - x^2 - 7x - 12 is divided by x - 2. We can substitute x = 2 into f(x):
f(2) = 2(2)^5 + 6(2)^4 - (2)^2 - 7(2) - 12
= 2(32) + 6(16) - 4 - 14 - 12
= 64 + 96 - 4 - 14 - 12
= 134.
Therefore, the remainder when f(x) is divided by x - 2 is 134.
To determine whether x - 2 is a factor of f(x), we can use the factor theorem. The factor theorem states that if a polynomial f(x) has a factor of x - a, then f(a) = 0.
In this case, to check if x - 2 is a factor of f(x), we need to substitute x = 2 into f(x) and see if the result is equal to zero.
f(2) = 2(2)^5 + 6(2)^4 - (2)^2 - 7(2) - 12
= 2(32) + 6(16) - 4 - 14 - 12
= 64 + 96 - 4 - 14 - 12
= 134.
Since f(2) is not equal to zero, x - 2 is not a factor of f(x).