Identify all real and non-real zeros of the function [tex]f(x)=x^3+5 x^2+3 x+15[/tex].
A. [tex]x=-5,1.7 i,-1.7 i[/tex]
B. [tex]x=0,-5,1.7 i[/tex]
C. [tex]x=0,-5,1.7 i,-1.7 i[/tex]
D. [tex]x=0,-3,-5[/tex]
Answer (2)
Factor the polynomial f ( x ) = x 3 + 5 x 2 + 3 x + 15 by grouping: f ( x ) = ( x 2 + 3 ) ( x + 5 ) .
Set each factor equal to zero: x + 5 = 0 and x 2 + 3 = 0 .
Solve for x : x = − 5 and x = ± i 3 .
The zeros are x = − 5 , ± i 3 , which are approximately x = − 5 , 1.732 i , − 1.732 i . Thus, the answer is x = − 5 , 1.732 i , − 1.732 i .
Explanation
Problem Analysis We are given the function f ( x ) = x 3 + 5 x 2 + 3 x + 15 and asked to find all its real and non-real zeros. This means we need to find all values of x for which f ( x ) = 0 .
Factoring the Polynomial To find the zeros, we can try to factor the polynomial. Factoring by grouping seems like a good approach here. We can group the first two terms and the last two terms: f ( x ) = ( x 3 + 5 x 2 ) + ( 3 x + 15 ) Now, we factor out the greatest common factor from each group: f ( x ) = x 2 ( x + 5 ) + 3 ( x + 5 ) We can see that ( x + 5 ) is a common factor, so we factor it out: f ( x ) = ( x 2 + 3 ) ( x + 5 )
Finding the Zeros Now we set f ( x ) = 0 to find the zeros: ( x 2 + 3 ) ( x + 5 ) = 0 This equation is satisfied if either x 2 + 3 = 0 or x + 5 = 0 .
Solving for x If x + 5 = 0 , then x = − 5 . This is a real zero. If x 2 + 3 = 0 , then x 2 = − 3 . Taking the square root of both sides, we get x = ± − 3 = ± i 3 . These are non-real zeros. Since 3 ≈ 1.732 , the non-real zeros are approximately x = 1.732 i and x = − 1.732 i .
Final Answer Therefore, the zeros of the function are x = − 5 , i 3 , − i 3 , which are approximately x = − 5 , 1.732 i , − 1.732 i .
Examples
Finding the zeros of a polynomial function is a fundamental concept in algebra and calculus. In real-world applications, this can be used to model various phenomena, such as the trajectory of a projectile, the behavior of electrical circuits, or the stability of structures. For example, engineers might use polynomial functions to model the stress on a bridge and find the points where the stress is zero, indicating potential weak points. Similarly, economists might use polynomial functions to model market trends and find the equilibrium points where supply equals demand. These equilibrium points are the zeros of the polynomial function representing the difference between supply and demand.
The zeros of the function f ( x ) = x 3 + 5 x 2 + 3 x + 15 are real zero x = − 5 and non-real zeros x = 1.732 i and x = − 1.732 i . Thus, the correct choice is A : x = − 5 , 1.732 i , − 1.732 i .
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