Use the Factor Theorem to determine whether x+2 is a factor of P(x)=2x^4+x^3-3x-9.
Specifically, evaluate P at the proper value, and then determine whether x+2 is a factor.
P(__)=
x+2 is or is not a factor of P(x)____
The factor theorem states that a polynomial f(x) has a factor (x − k) if and only if f(k) = 0.
So if (x+2) is a factor of P(x), then
P(-2)=0.
Since P(-2)= 21, it is clear that (x+2) is not a factor of P(x)=2x^4+x^3-3x-9.
Well, let's plug in -2 for x and see what happens.
P(-2) = 2(-2)^4 + (-2)^3 - 3(-2) - 9
= 2(16) + (-8) + 6 - 9
= 32 - 8 + 6 - 9
= 31
So, P(-2) = 31.
Now, to determine whether x+2 is a factor, we need to check if P(-2) is equal to zero.
Since P(-2) is not equal to zero (it's equal to 31), we can conclude that x+2 is not a factor of P(x).
But hey, at least P(-2) made us laugh with that unexpected answer!
To determine whether x+2 is a factor of P(x), we need to evaluate P(-2) and check if the result is equal to zero.
To evaluate P(-2), substitute -2 for x in the expression P(x):
P(-2) = 2(-2)^4 + (-2)^3 - 3(-2) - 9
Simplifying this expression:
P(-2) = 2(16) + (-8) + 6 - 9
P(-2) = 32 - 8 + 6 - 9
P(-2) = 21
Since P(-2) is not equal to zero, x+2 is not a factor of P(x).
To use the Factor Theorem to determine whether x + 2 is a factor of P(x) = 2x^4 + x^3 - 3x - 9, we need to evaluate P(x) at the value that makes x + 2 equal to zero.
Setting x + 2 = 0 and solving for x, we find that x = -2.
Now, substitute x = -2 into P(x) to find P(-2):
P(-2) = 2(-2)^4 + (-2)^3 - 3(-2) - 9
= 32 - 8 + 6 - 9
= 21
Since P(-2) does not equal zero, x + 2 is not a factor of P(x).
Therefore, P(x) = 2x^4 + x^3 - 3x - 9
x + 2 is not a factor of P(x).