Answer:
x = (3 ± sqrt(-47)) / 2.
Explanation:
If x - 6 is a factor of f(x), then we know that (x - 6) must divide evenly into f(x), which means that there is another factor of f(x) that we can find by polynomial long division or synthetic division.
Using synthetic division:
We start by writing the coefficients of f(x) in order:
1 3 -22 24
We then write the factor (x - 6) to the left of the coefficients, and draw a line:
6 | 1 3 -22 24
We bring down the first coefficient: 1
1 3 -22 24
1
We multiply 6 by the first coefficient and write the result under the second coefficient: 1
6 | 1 3 -22 24
-6
-3
If x - 6 is a factor of f(x), then we know that (x - 6) must divide evenly into f(x), which means that there is another factor of f(x) that we can find by polynomial long division or synthetic division.
Using synthetic division:
We start by writing the coefficients of f(x) in order:
1 3 -22 24
We then write the factor (x - 6) to the left of the coefficients, and draw a line:
6 | 1 3 -22 24
We bring down the first coefficient:
Copy code
1
6 | 1 3 -22 24
1
We multiply 6 by the first coefficient and write the result under the second coefficient:
Copy code
1
6 | 1 3 -22 24
-6
Copy code
-3
We add the second and third coefficients to get -19, then multiply by 6 and write the result under the third coefficient:
1
6 | 1 3 -22 24
-6
-3
42
66
We add the last two numbers to get 90. This means that we can write f(x) as:
f(x) = (x - 6)(x² - 3x + 14)
To find the zeros of f(x), we need to solve the equation (x - 6)(x² - 3x + 14) = 0.
The first factor gives us x = 6.
To solve the quadratic factor, we can use the quadratic formula:
x = (-b ± sqrt(b² - 4ac)) / 2a
In this case, a = 1, b = -3, and c = 14. Substituting these values into the formula, we get:
x = (3 ± sqrt(3² - 4(1)(14))) / 2(1)
x = (3 ± sqrt(-47)) / 2
Since the square root of a negative number is not a real number, the quadratic factor does not have any real zeros.
Therefore, the zeros of f(x) are x = 6, and the complex numbers x = (3 ± sqrt(-47)) / 2.