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)
They told you that x = -4 was a root so (x+4) is a factor so divide 3x^3 + 10x^2 - 16x - 32 by (x+4)
I get
3 x^2 - 2 x - 8
Factor that
3 * 1 factors 3
8*1 or 2*4 factors 8
so try 2 and 4
(3x -2)(x+4), nope
(3x-4)(x+2) close
(3x+4)(x-2) aha
so all the roots are
-4
-4/3
+2
To solve this problem, we will follow these steps:
a) Finding the depressed equation:
1. Write down the original equation: f(x) = 3x^3 + 10x^2 - 16x - 32.
2. Use synthetic division to divide the original equation by the root given (x = -4). Synthetic division works by reducing the degree of the polynomial by one.
-4 | 3 10 -16 -32
|_____ -12 8 32
3 -2 -8 0
3. The result of synthetic division is a quotient and a remainder, with the coefficients of the depressed equation on the top line: f(x) = 3x^2 - 2x - 8.
b) Factoring the depressed equation:
To factor the quadratic equation f(x) = 3x^2 - 2x - 8, we can use any factorization method like factoring by grouping, completing the square, or the quadratic formula. Let's use factoring by grouping:
1. Split the middle term (-2x) into two terms whose coefficients multiply to give the product of the first and last coefficients (3 * -8 = -24) and add up to the middle coefficient (-2).
3x^2 - 6x + 4x - 8
2. Factor by grouping by grouping the pairs of terms:
3x(x - 2) + 4(x - 2)
3. Notice that we have a common factor (x - 2) in both terms. Factor out this common factor:
(x - 2)(3x + 4)
So, the factored form of the depressed equation is f(x) = (x - 2)(3x + 4).
c) Listing all the roots for f(x):
To find all the roots of f(x), we set each factor equal to zero and solve for x:
1. Set (x - 2) = 0, and solve for x:
x - 2 = 0
x = 2
2. Set (3x + 4) = 0, and solve for x:
3x + 4 = 0
3x = -4
x = -4/3
Therefore, the roots of the polynomial f(x) = 3x^3 + 10x^2 - 16x - 32 are x = 2, and x = -4/3.