To determine if a given expression is a factor of f(x), we need to check if f(x) is divisible by that expression.
Let's check each expression individually:
1. (x+7):
To determine if (x+7) is a factor of f(x), we need to check if f(-7) equals zero.
When we plug in x=-7 into f(x), we get:
f(-7) = 2(-7)^3 - 5(-7)^2 - 68(-7) + 35
= 2(-343) - 5(49) + 476 + 35
= -686 - 245 + 476 + 35
= -686 - 210 + 511
= -896 + 511
= -385
Since f(-7) does not equal zero, (x+7) is not a factor of f(x).
2. (x-3):
To determine if (x-3) is a factor of f(x), we need to check if f(3) equals zero.
When we plug in x=3 into f(x), we get:
f(3) = 2(3)^3 - 5(3)^2 - 68(3) + 35
= 2(27) - 5(9) - 204 + 35
= 54 - 45 - 204 + 35
= 9 - 204 + 35
= -195 + 35
= -160
Since f(3) does not equal zero, (x-3) is not a factor of f(x).
3. (x+5):
To determine if (x+5) is a factor of f(x), we need to check if f(-5) equals zero.
When we plug in x=-5 into f(x), we get:
f(-5) = 2(-5)^3 - 5(-5)^2 - 68(-5) + 35
= 2(-125) - 5(25) + 340 + 35
= -250 - 125 + 340 + 35
= -375 + 375
= 0
Since f(-5) equals zero, (x+5) is a factor of f(x).
4. (2x+1):
To determine if (2x+1) is a factor of f(x), we need to check if f(-1/2) equals zero.
When we plug in x=-1/2 into f(x), we get:
f(-1/2) = 2(-1/2)^3 - 5(-1/2)^2 - 68(-1/2) + 35
= 2(-1/8) - 5(1/4) + 34 + 35
= -1/4 - 5/4 + 34 + 35
= -6/4 + 34 + 35
= -3/2 + 34 + 35
= 31/2 + 34
Since f(-1/2) does not equal zero, (2x+1) is not a factor of f(x).
Therefore, the only expression that is a factor of f(x) is (x+5).
Given f(x)=2x^3−5x^2−68x+35, apply the Factor Theorem and determine which of the following is a factor of f(x).
(x+7)
(x−3)
(x+5)
(2x+1)
1 answer