Find the zeros of [tex]f(x)=x^3+6 x^2+8 x[/tex].
A) [tex]x=2,4[/tex]
B) [tex]x=0,2,4[/tex]
C) [tex]x=0,-2,-4[/tex]
D) [tex]x=-2,-4[/tex]

Factor out the common factor x from the polynomial: f ( x ) = x ( x 2 + 6 x + 8 ) .
Factor the quadratic expression: x 2 + 6 x + 8 = ( x + 2 ) ( x + 4 ) .
Set each factor to zero and solve for x : x = 0 , x + 2 = 0 ⇒ x = − 2 , x + 4 = 0 ⇒ x = − 4 .
The zeros of the function are x = 0 , − 2 , − 4 ​ .

Explanation

Understanding the Problem We are asked to find the zeros of the function f ( x ) = x 3 + 6 x 2 + 8 x . This means we need to find the values of x for which f ( x ) = 0 .

Factoring the Polynomial First, we can factor out a common factor of x from the expression: f ( x ) = x ( x 2 + 6 x + 8 ) Now we need to find the values of x that make this expression equal to zero.

Factoring the Quadratic We can further factor the quadratic expression x 2 + 6 x + 8 . We are looking for two numbers that multiply to 8 and add to 6. These numbers are 2 and 4. So we can write the quadratic as: x 2 + 6 x + 8 = ( x + 2 ) ( x + 4 ) Thus, the factored form of the function is: f ( x ) = x ( x + 2 ) ( x + 4 )

Finding the Zeros Now we set each factor equal to zero and solve for x :



x = 0
x + 2 = 0 ⇒ x = − 2
x + 4 = 0 ⇒ x = − 4

So the zeros of the function are x = 0 , − 2 , − 4 .

Final Answer The zeros of the function f ( x ) = x 3 + 6 x 2 + 8 x are 0 , − 2 , and − 4 . Therefore, the correct answer is C) x = 0 , − 2 , − 4 .

Examples
Finding the zeros of a polynomial can help us understand the behavior of systems modeled by polynomials. For example, in engineering, the stability of a system can be determined by finding the roots of its characteristic equation, which is often a polynomial. If the roots have negative real parts, the system is stable; otherwise, it is unstable. This concept is crucial in designing control systems, signal processing, and many other applications.

Answered by GinnyAnswer | 2025-07-07