Identify the zeros of [tex]f(x)=x^3+4 x^2-9 x-36[/tex]
A. [tex]x=-9,-4[/tex]
B. [tex]x=9,-4[/tex]
C. [tex]x=-3,3,4[/tex]
D. [tex]x=-3,3,-4[/tex]
Answer (2)
Factor the polynomial f ( x ) = x 3 + 4 x 2 − 9 x − 36 by grouping: f ( x ) = ( x + 4 ) ( x 2 − 9 ) .
Factor the difference of squares: f ( x ) = ( x + 4 ) ( x − 3 ) ( x + 3 ) .
Set each factor to zero: x + 4 = 0 , x − 3 = 0 , x + 3 = 0 .
Solve for x to find the zeros: x = − 3 , 3 , − 4 .
Explanation
Understanding the Problem We are given the function f ( x ) = x 3 + 4 x 2 − 9 x − 36 and asked to find its zeros. The zeros of a function are the values of x for which f ( x ) = 0 . In other words, we need to solve the equation x 3 + 4 x 2 − 9 x − 36 = 0 .
Factoring by Grouping To solve this equation, we can try factoring the polynomial. We can use factoring by grouping. We group the first two terms and the last two terms: ( x 3 + 4 x 2 ) + ( − 9 x − 36 ) . From the first group, we can factor out x 2 , and from the second group, we can factor out − 9 : x 2 ( x + 4 ) − 9 ( x + 4 ) . Now we see that ( x + 4 ) is a common factor, so we can factor it out: ( x + 4 ) ( x 2 − 9 ) = 0 .
Factoring Difference of Squares Now we can further factor the term ( x 2 − 9 ) , which is a difference of squares. We have x 2 − 9 = ( x − 3 ) ( x + 3 ) . So the equation becomes ( x + 4 ) ( x − 3 ) ( x + 3 ) = 0 .
Finding the Zeros To find the zeros, we set each factor equal to zero and solve for x :
x + 4 = 0 ⇒ x = − 4
x − 3 = 0 ⇒ x = 3
x + 3 = 0 ⇒ x = − 3
Thus, the zeros of the function are x = − 4 , 3 , − 3 .
Examples
Finding the zeros of a polynomial is a fundamental concept in algebra and calculus. For example, if you were designing a suspension bridge, you might model the sag of the suspension cable with a polynomial function. The zeros of this function would represent the points where the cable touches the ground (or the supports). Knowing these points is crucial for the structural integrity and safety of the bridge. Similarly, in economics, finding the roots of a cost function can help determine the break-even points for a business.
The zeros of the polynomial function f ( x ) = x 3 + 4 x 2 − 9 x − 36 are found to be x = − 4 , 3 , − 3 . This corresponds to option D: x = − 3 , 3 , − 4 .
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