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).
1. (x-3)
2. (2x+1)
3. (x+5)
4. (x+7)
To apply the Factor Theorem, we need to check if any of the options given are roots of the polynomial f(x).
Let's check each option.
1. (x-3): If we substitute x=3 into f(x), we get f(3) = 2(3)^3 - 5(3)^2 - 68(3) + 35 = 54 - 45 - 204 + 35 = -160. Since f(3) is not equal to zero, (x-3) is not a factor of f(x).
2. (2x+1): If we substitute x=-1/2 into f(x), we get f(-1/2) = 2(-1/2)^3 - 5(-1/2)^2 - 68(-1/2) + 35 = -1/4 - 5/4 + 34 + 35 = 63/4. Since f(-1/2) is not equal to zero, (2x+1) is not a factor of f(x).
3. (x+5): If we substitute x=-5 into f(x), we get f(-5) = 2(-5)^3 - 5(-5)^2 - 68(-5) + 35 = -250 + 125 + 340 + 35 = 250. Since f(-5) is not equal to zero, (x+5) is not a factor of f(x).
4. (x+7): If we substitute x=-7 into f(x), we get f(-7) = 2(-7)^3 - 5(-7)^2 - 68(-7) + 35 = -686 + 245 - 476 + 35 = -882. Since f(-7) is not equal to zero, (x+7) is not a factor of f(x).
None of the options given (1, 2, 3, 4) are factors of f(x).
To determine if a polynomial is a factor of another polynomial, we can use the Factor Theorem. According to the Factor Theorem, if a polynomial f(x) has a factor (x-a), then f(a) will equal to zero.
Let's apply the Factor Theorem to each of the given options:
1. (x-3):
To check if (x-3) is a factor, we need to evaluate f(3):
f(3) = 2(3)^3 - 5(3)^2 - 68(3) + 35
= 54 - 45 - 204 + 35
= -160
Since f(3) is not equal to zero, (x-3) is not a factor of f(x).
2. (2x+1):
To check if (2x+1) is a factor, we need to evaluate f(-1/2):
f(-1/2) = 2(-1/2)^3 - 5(-1/2)^2 - 68(-1/2) + 35
= -1/4 - 5/4 + 34 + 35
= 67
Since f(-1/2) is not equal to zero, (2x+1) is not a factor of f(x).
3. (x+5):
To check if (x+5) is a factor, we need to evaluate f(-5):
f(-5) = 2(-5)^3 - 5(-5)^2 - 68(-5) + 35
= -250 - 125 + 340 + 35
= 0
Since f(-5) is equal to zero, (x+5) is a factor of f(x).
4. (x+7):
To check if (x+7) is a factor, we need to evaluate f(-7):
f(-7) = 2(-7)^3 - 5(-7)^2 - 68(-7) + 35
= -686 + 245 + 476 + 35
= 70
Since f(-7) is not equal to zero, (x+7) is not a factor of f(x).
Therefore, the correct answer is option 3. (x+5) is a factor of f(x).
To determine if a given polynomial is a factor of f(x), we can utilize the Factor Theorem. According to the Factor Theorem, a polynomial p(x) is a factor of another polynomial f(x) if and only if p(a) = 0, where a is a root of f(x).
Let's apply the Factor Theorem to each of the given options and check if any of them are factors of f(x):
1. (x-3):
To determine if (x-3) is a factor of f(x), we need to check if f(3) = 0. Let's substitute x = 3 into f(x) and evaluate:
f(3) = 2(3)^3 - 5(3)^2 - 68(3) + 35
= 2(27) - 5(9) - 204 + 35
= 54 - 45 - 204 + 35
= -160
Since f(3) = -160 is not equal to 0, (x-3) is not a factor of f(x).
2. (2x+1):
To determine if (2x+1) is a factor of f(x), we need to check if f(-1/2) = 0. Let's substitute x = -1/2 into f(x) and evaluate:
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
= 29 - 5/4
Since f(-1/2) ≠ 0, (2x+1) 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) = 0. Let's substitute x = -5 into f(x) and evaluate:
f(-5) = 2(-5)^3 - 5(-5)^2 - 68(-5) + 35
= 2(-125) - 5(25) + 340 + 35
= -250 - 125 + 340 + 35
= 0
Since f(-5) = 0, (x+5) is a factor of f(x).
4. (x+7):
To determine if (x+7) is a factor of f(x), we need to check if f(-7) = 0. Let's substitute x = -7 into f(x) and evaluate:
f(-7) = 2(-7)^3 - 5(-7)^2 - 68(-7) + 35
= 2(-343) - 5(49) + 476 + 35
= -686 - 245 + 476 + 35
= -420
Since f(-7) ≠ 0, (x+7) is not a factor of f(x).
Therefore, the correct answer is option 3. (x+5) is a factor of f(x).