Sketch the following polynomial function using the four-step process.
[tex]f(x)=-x^5+6 x^4+36 x^3-216 x^2[/tex]
The left-hand behavior starts $\square$ up and the right-hand behavior ends down.
Find the [tex]y[/tex]-intercept.
The [tex]y[/tex]-intercept is [tex]y=[/tex] $\square$ 0.
The real zeros of the polynomial are [tex]x =[/tex] $\square$.
(Use a comma to separate answers as needed. Type an exact answer, using radicals as needed.)
Answer (2)
Factor the polynomial: f ( x ) = − x 2 ( x − 6 ) 2 ( x + 6 ) .
Identify the zeros and their multiplicities: x = − 6 (multiplicity 1), x = 0 (multiplicity 2), x = 6 (multiplicity 2).
Determine the end behavior: left-hand starts up, right-hand ends down.
State the zeros: x = − 6 , 0 , 6
Explanation
Problem Analysis Let's analyze the given polynomial function: f ( x ) = − x 5 + 6 x 4 + 36 x 3 − 216 x 2 . Our goal is to sketch this polynomial by finding its zeros, y-intercept, and describing its end behavior.
Factoring the Polynomial First, we can factor the polynomial to find its zeros:
f ( x ) = − x 5 + 6 x 4 + 36 x 3 − 216 x 2 = − x 2 ( x 3 − 6 x 2 − 36 x + 216 )
Now, let's factor the cubic expression x 3 − 6 x 2 − 36 x + 216 . We can use factoring by grouping:
x 3 − 6 x 2 − 36 x + 216 = x 2 ( x − 6 ) − 36 ( x − 6 ) = ( x 2 − 36 ) ( x − 6 ) = ( x − 6 ) ( x + 6 ) ( x − 6 ) = ( x − 6 ) 2 ( x + 6 )
So, the factored form of the polynomial is:
f ( x ) = − x 2 ( x − 6 ) 2 ( x + 6 )
Finding the Zeros Now, we can identify the zeros of the polynomial and their multiplicities:
x = 0 with multiplicity 2
x = 6 with multiplicity 2
x = − 6 with multiplicity 1
Finding the y-intercept Next, let's find the y-intercept by setting x = 0 in the polynomial:
f ( 0 ) = − 0 5 + 6 ( 0 ) 4 + 36 ( 0 ) 3 − 216 ( 0 ) 2 = 0
So, the y-intercept is y = 0 .
Determining End Behavior Now, let's determine the end behavior of the polynomial. The leading term is − x 5 . Since the degree is odd and the leading coefficient is negative, the left-hand behavior starts up and the right-hand behavior ends down.
Sketching the Polynomial Finally, we can sketch the polynomial using the zeros, their multiplicities, the y-intercept, and the end behavior. The zeros are x = − 6 , 0 , 6 . At x = − 6 , the graph crosses the x-axis since the multiplicity is 1. At x = 0 and x = 6 , the graph touches the x-axis and turns around since the multiplicity is 2.
Final Answer The left-hand behavior starts up and the right-hand behavior ends down. The y-intercept is y = 0 . The real zeros of the polynomial are x = − 6 , 0 , 6 .
Examples
Polynomial functions are used in various real-world applications, such as modeling curves in engineering and physics. For example, the trajectory of a projectile can be modeled using a polynomial function. Understanding the zeros and end behavior of a polynomial helps engineers design structures and predict the behavior of systems.
The polynomial f ( x ) = − x 5 + 6 x 4 + 36 x 3 − 216 x 2 factors to give the zeros at x = − 6 , 0 , 6 with their respective multiplicities. The y-intercept is y = 0 , and the end behavior shows that it starts up on the left and ends down on the right. Therefore, the left behavior is up, the right is down, with real zeros at the specified values.
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