Identify the zeros of [tex]f(x)=2 x^6+2 x^5[/tex].
A. [tex]x=0,-1,1[/tex]
B. [tex]x=0,1[/tex]
C. [tex]x=0,-1[/tex]
D. [tex]x=0,-2[/tex]
Answer (2)
Set the function equal to zero: 2 x 6 + 2 x 5 = 0 .
Factor out the common term: 2 x 5 ( x + 1 ) = 0 .
Solve for x by setting each factor to zero: 2 x 5 = 0 and x + 1 = 0 .
The zeros of the function are: x = 0 , − 1 .
Explanation
Understanding the Problem We are asked to find the zeros of the function f ( x ) = 2 x 6 + 2 x 5 . This means we need to find 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 : 2 x 6 + 2 x 5 = 0
Factoring the Expression We can factor out the common term 2 x 5 from the expression: 2 x 5 ( x + 1 ) = 0
Solving the First Factor Now, we set each factor equal to zero and solve for x . First, we have: 2 x 5 = 0 Dividing both sides by 2, we get: x 5 = 0 Taking the fifth root of both sides, we find: x = 0
Solving the Second Factor Next, we have: x + 1 = 0 Subtracting 1 from both sides, we get: x = − 1
Final Answer Therefore, the zeros of the function f ( x ) = 2 x 6 + 2 x 5 are x = 0 and x = − 1 .
Examples
Understanding the zeros of a function is crucial in many real-world applications. For example, in physics, the zeros of a function might represent the points where a projectile hits the ground. In engineering, they could represent the equilibrium points of a system. In economics, they might represent the break-even points for a business. By finding the zeros, we can analyze and predict the behavior of these systems.
The zeros of the function f ( x ) = 2 x 6 + 2 x 5 are x = 0 and x = − 1 . Therefore, the correct answer is option A: x = 0 , − 1 .
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