Lesson 4
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Identify the zeros of [tex]f(x)=x^3+3 x^2+2 x[/tex]
A. [tex]x=-1,-2[/tex]
B. [tex]x=1,2[/tex]
C. [tex]x=0,-1,-2[/tex]
D. [tex]x=0,1,2[/tex]
Answer (1)
Set the function equal to zero: x 3 + 3 x 2 + 2 x = 0 .
Factor out x : x ( x 2 + 3 x + 2 ) = 0 .
Factor the quadratic: x ( x + 1 ) ( x + 2 ) = 0 .
Solve for x : The zeros are x = 0 , − 1 , − 2 .
Explanation
Understanding the Problem We are given the function f ( x ) = x 3 + 3 x 2 + 2 x and asked to find its zeros. The zeros of a function are the values of x for which f ( x ) = 0 .
Setting up the Equation To find the zeros, we set f ( x ) = 0 and solve for x :
x 3 + 3 x 2 + 2 x = 0
Factoring out x We can factor out an x from each term: x ( x 2 + 3 x + 2 ) = 0
Factoring the Quadratic Now we need to factor the quadratic expression x 2 + 3 x + 2 . We are looking for two numbers that multiply to 2 and add to 3. These numbers are 1 and 2. So we can factor the quadratic as: ( x + 1 ) ( x + 2 )
Complete Factored Form Substituting this back into our equation, we have: x ( x + 1 ) ( x + 2 ) = 0
Finding the Zeros The zeros of the function are the values of x that make this equation true. This occurs when x = 0 , x + 1 = 0 , or x + 2 = 0 . Solving these equations, we get:
x = 0 x + 1 = 0 ⇒ x = − 1 x + 2 = 0 ⇒ x = − 2
So the zeros are x = 0 , − 1 , − 2 .
Final Answer Therefore, the zeros of the function f ( x ) = x 3 + 3 x 2 + 2 x are 0 , − 1 , and − 2 .
Examples
Finding the zeros of a polynomial is a fundamental concept in algebra and calculus. In real-world applications, finding zeros is essential in various fields such as engineering, physics, and economics. For example, in engineering, zeros can represent the points where a structure experiences no stress or strain. In physics, they can represent equilibrium points in a system. In economics, they can represent break-even points where cost equals revenue. Understanding how to find zeros allows professionals to analyze and design systems effectively.
