determine the cubic equation that has zeros at -8,-4and 13 if f(-10) =-138.
To determine the cubic equation that has zeros at -8, -4, and 13, we can use the factored form of a cubic equation.
The factored form of a cubic equation with zeros at a, b, and c is given by:
f(x) = k(x - a)(x - b)(x - c)
In this case, the zeros are -8, -4, and 13, so the equation becomes:
f(x) = k(x + 8)(x + 4)(x - 13)
Now, we need to find the value of k to satisfy f(-10) = -138.
Substituting x = -10 into the equation, we have:
-138 = k(-10 + 8)(-10 + 4)(-10 - 13)
Simplifying this equation, we get:
-138 = k(-2)(-6)(-23)
To solve for k, we divide both sides of the equation by (-2)(-6)(-23):
k = -138 / (2 * 6 * 23)
Simplifying, we find:
k = -3/23
Therefore, the cubic equation is:
f(x) = (-3/23)(x + 8)(x + 4)(x - 13)
To determine the cubic equation with the given zeros and the value of f(-10), we can use the fact that if r is a root of a polynomial, then (x-r) is a factor of that polynomial.
Step 1: Determine the factors
Since the zeros are -8, -4, and 13, we have the factors (x+8), (x+4), and (x-13).
Step 2: Form the equation
To get the cubic equation, we can multiply the factors together:
(x+8)(x+4)(x-13)
Step 3: Simplify
Expand the equation:
(x+8)(x+4)(x-13) = (x^2 + 12x + 32)(x-13)
= x^3 + 12x^2 + 32x - 13x^2 - 156x - 416
= x^3 - x^2 - 124x - 416
Therefore, the cubic equation with zeros at -8, -4, and 13 is:
f(x) = x^3 - x^2 - 124x - 416
To check if f(-10) equals -138, substitute -10 into the equation:
f(-10) = (-10)^3 - (-10)^2 - 124(-10) - 416
= -1000 - 100 + 1240 - 416
= -276
Since f(-10) does not equal -138, it seems there might be a mistake in the given information or solution.
knowing the roots, we have
y(x) = a(x+8)(x+4)(x-13)
Now plug in y(-10) = -138 to find what a is.