The equation f(x) = 3x^3 + 10x^2 - 16x - 32 has a root at x = -4
a) Find the depressed equation using synthetic division and the root given. Factor out any constants that may appear.
b) Factor the depressed equation (which is now a quadratic) using any method EXCEPT synthetic division.
c) List all the roots for f(x)
You get the "depressed equation" by dividing
3x^3 + 10x^2 - 16x - 32 by the factor already. This is a skil you need to learn.
x + 4. Read up on how to do synthetic division if you don't know already. Show your work if you need additional help.
The depressed quadratic equation (if f(x) - 0) is
3x^2 -2x -8 = 0
That does not factor easily
a) To find the depressed equation using synthetic division and the given root at x = -4, we will divide the original equation by the linear factor (x - (-4)) or (x + 4):
First, write the coefficients of the equation: 3, 10, -16, -32.
Next, arrange the coefficients in descending order: 3, 10, -16, -32.
Now, set up the synthetic division as follows:
-4 | 3 10 -16 -32
|_______
Bring down the first coefficient, which is 3:
-4 | 3 10 -16 -32
|_______
3
Multiply the root, -4, by the previous result (3) and write the product below the next coefficient:
-4 | 3 10 -16 -32
|_______
3
-12
Add the new value to the next coefficient (10):
-4 | 3 10 -16 -32
|_______
3 -2
-12
Repeat the process:
-4 | 3 10 -16 -32
|_______
3 -2 36
-12 8
The final line of numbers represents the coefficients of the depressed equation:
3x^2 - 2x + 8
b) To factor the depressed equation (3x^2 - 2x + 8), we can use the quadratic factoring method. However, it is important to note that the quadratic equation does not factor into linear factors since the discriminant (√(b^2 - 4ac)) is negative.
To factor the quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For the equation 3x^2 - 2x + 8:
a = 3
b = -2
c = 8
Using the quadratic formula to find the roots:
x = (-(-2) ± √((-2)^2 - 4(3)(8))) / (2(3))
= (2 ± √(4 - 96)) / 6
= (2 ± √(-92)) / 6
Since the discriminant (√(-92)) is negative, the quadratic equation does not have real roots. The factors of the depressed equation are complex numbers, which can be written in the form:
x = (2 ± (i√92)) / 6
= (1 ± (i√23)) / 3
c) To list all the roots for f(x), we combine the root found in part a (x = -4) with the complex roots found in part b:
The roots for f(x) are:
1) x = -4 (real root)
2) x = (1 + i√23) / 3 (complex root)
3) x = (1 - i√23) / 3 (complex root)