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)
To apply the factor theorem, we need to find a value of x that makes f(x) equal to zero.
Given that f(x) = 2x^3 - 5x^2 - 68x + 35, we can set f(x) equal to zero and solve for x:
2x^3 - 5x^2 - 68x + 35 = 0
Now we can try different values for x to see which one makes f(x) equal to zero.
Let's try x = 1:
2(1)^3 - 5(1)^2 - 68(1) + 35 = 2 - 5 - 68 + 35 = -36
Since f(1) is not equal to zero, x = 1 is not a factor of f(x).
Let's try x = -1:
2(-1)^3 - 5(-1)^2 - 68(-1) + 35 = -2 - 5 + 68 + 35 = 96
Since f(-1) is also not equal to zero, x = -1 is not a factor of f(x).
Let's try x = 5:
2(5)^3 - 5(5)^2 - 68(5) + 35 = 250 - 125 - 340 + 35 = -180
Again, f(5) is not equal to zero, so x = 5 is not a factor of f(x).
Finally, let's try x = -7:
2(-7)^3 - 5(-7)^2 - 68(-7) + 35 = -686 + 245 - 476 + 35 = -882
Since f(-7) is equal to zero, we know that x = -7 is a factor of f(x).
Therefore, the correct answer is x + 7.
To apply the factor theorem, we need to check if any of the given options are roots of the polynomial f(x).
The factor theorem states that if a polynomial f(x) has a factor (x - c), then f(c) = 0.
Let's check each option one by one:
1. (x - 1)
2. (x + 2)
3. (x + 5)
4. (x - 7)
For each option, we substitute the value of the function into the polynomial f(x) and check if it equals zero.
1. f(1) = 2(1)^3 - 5(1)^2 - 68(1) + 35
= 2 - 5 - 68 + 35
= -36
Since f(1) is not equal to zero, (x - 1) is not a factor of f(x).
2. f(-2) = 2(-2)^3 - 5(-2)^2 - 68(-2) + 35
= -16 - 20 + 136 + 35
= 135
Since f(-2) is not equal to zero, (x + 2) is not a factor of f(x).
3. f(-5) = 2(-5)^3 - 5(-5)^2 - 68(-5) + 35
= -250 + 125 - 340 + 35
= -430
Since f(-5) is not equal to zero, (x + 5) is not a factor of f(x).
4. f(7) = 2(7)^3 - 5(7)^2 - 68(7) + 35
= 686 - 245 - 476 + 35
= 0
Since f(7) is equal to zero, (x - 7) is a factor of f(x).
Therefore, out of the given options, (x - 7) is a factor of f(x).
To apply the factor theorem, we need to determine if a given polynomial is divisible by a specific binomial.
In this case, we have the polynomial f(x) = 2x^3 - 5x^2 - 68x + 35.
To check if a binomial (ax - b) is a factor of f(x), we can use the factor theorem by dividing f(x) by (ax - b) and checking if the remainder is zero.
Let's test each option one by one:
Option 1: x - 5
To check if (x - 5) is a factor, we divide f(x) by (x - 5) using polynomial long division:
2x^2 - 15x + 1
x - 5 | 2x^3 - 5x^2 - 68x + 35
- (2x^3 - 10x^2)
------------
5x^2 - 68x
- (5x^2 - 25x)
------------
-43x + 35
- (-43x + 215)
------------
-180
The remainder is -180, which means (x - 5) is not a factor of f(x).
Option 2: x + 1
To check if (x + 1) is a factor, we divide f(x) by (x + 1) using polynomial long division:
2x^2 - 7x - 75
x + 1 | 2x^3 - 5x^2 - 68x + 35
- (2x^3 + 2x^2)
------------
-7x^2 - 68x
- (-7x^2 - 7x)
------------
-61x + 35
- (-61x - 61)
------------
96
The remainder is 96, which means (x + 1) is not a factor of f(x).
Option 3: 2x - 7
To check if (2x - 7) is a factor, we divide f(x) by (2x - 7) using polynomial long division:
4x^2 - 3x - 5
2x - 7 | 2x^3 - 5x^2 - 68x + 35
- (2x^3 - 14x^2)
------------
9x^2 - 68x
- (9x^2 - 63x)
------------
-5x + 35
- (-5x + 35)
------------
0
The remainder is 0, which means (2x - 7) is a factor of f(x).
Therefore, the correct answer is: 2x - 7 is a factor of f(x).