To find the zeros of the polynomial function [tex]\( f(x) = x^3 + 4x^2 - 9x - 36 \)[/tex], we need to determine the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 0 \)[/tex]. Here is the step-by-step process to solve for the zeros:
1. Define the Polynomial Equation:
[tex]\[
f(x) = x^3 + 4x^2 - 9x - 36
\][/tex]
2. Set the Polynomial Equal to Zero:
[tex]\[
x^3 + 4x^2 - 9x - 36 = 0
\][/tex]
3. Find the Zeros:
Solving this cubic equation exactly can sometimes be challenging and often requires various algebraic techniques such as factoring, graphing, or numerical methods. However, in this particular instance, the solutions for the equation are:
[tex]\[
x = -4, \quad x = -3, \quad x = 3
\][/tex]
4. Verify the Solutions:
Let's verify these by substituting them back into the original polynomial equation to check if they yield zero.
- For [tex]\( x = -4 \)[/tex]:
[tex]\[
(-4)^3 + 4(-4)^2 - 9(-4) - 36 = -64 + 64 + 36 - 36 = 0
\][/tex]
- For [tex]\( x = -3 \)[/tex]:
[tex]\[
(-3)^3 + 4(-3)^2 - 9(-3) - 36 = -27 + 36 + 27 - 36 = 0
\][/tex]
- For [tex]\( x = 3 \)[/tex]:
[tex]\[
3^3 + 4(3)^2 - 9(3) - 36 = 27 + 36 - 27 - 36 = 0
\][/tex]
Each substitution yields zero, confirming that [tex]\( x = -4 \)[/tex], [tex]\( x = -3 \)[/tex], and [tex]\( x = 3 \)[/tex] are indeed the zeros of the polynomial function.
Therefore, the zeros of the polynomial function [tex]\( f(x) = x^3 + 4x^2 - 9x - 36 \)[/tex] are:
[tex]\[ \boxed{-4, -3, 3} \][/tex]
