Find the remaining zeros of [tex]f(x)[/tex] given that [tex]c[/tex] is a zero. Then rewrite [tex]f(x)[/tex] in completely factored form and sketch its graph. [tex]f(x)=2 x^5+5 x^4+2 x^3-4 x^2-4 x-1 ; c=-1[/tex] is a zero of multiplicity three
Identify all the remaining zeros.
[tex]x=[/tex]
(Simplify your answer. Type an integer or a fraction. Use a comma to separate answers as needed.)
Write the completely factored form of [tex]f(x)[/tex].
[tex]f(x)=[/tex]
(Simplify your answer. Use integers or fractions for any numbers in the expression.)
Answer (2)
The polynomial f ( x ) has a zero at x = − 1 with multiplicity 3, and remaining zeros at x = 1 and x = − 2 1 .
Express f ( x ) as a ( x + 1 ) 3 ( x − 1 ) ( x + 2 1 ) and expand the expression.
Compare the expanded form with the original polynomial to find the constant a = 2 .
Write the completely factored form as f ( x ) = ( x + 1 ) 3 ( x − 1 ) ( 2 x + 1 ) .
f ( x ) = ( x + 1 ) 3 ( x − 1 ) ( 2 x + 1 )
Explanation
Problem Analysis We are given the polynomial f ( x ) = 2 x 5 + 5 x 4 + 2 x 3 − 4 x 2 − 4 x − 1 and told that c = − 1 is a zero of multiplicity three. This means that ( x + 1 ) 3 is a factor of f ( x ) . We are also given that the remaining zeros are x = 1 and x = − 2 1 . Therefore, ( x − 1 ) and ( x + 2 1 ) are also factors of f ( x ) . Our goal is to write f ( x ) in completely factored form.
Factoring with Unknown Constant Since we know all the zeros of f ( x ) , we can write it in the form f ( x ) = a ( x + 1 ) 3 ( x − 1 ) ( x + 2 1 ) for some constant a . To find the value of a , we can expand the expression and compare it to the original polynomial.
Expanding the Factors Expanding ( x + 1 ) 3 ( x − 1 ) ( x + 2 1 ) gives us: ( x 3 + 3 x 2 + 3 x + 1 ) ( x 2 − 2 1 x − 2 1 ) = x 5 + 3 x 4 + 3 x 3 + x 2 − 2 1 x 4 − 2 3 x 3 − 2 3 x 2 − 2 1 x − 2 1 x 3 − 2 3 x 2 − 2 3 x − 2 1 = x 5 + 2 5 x 4 + x 3 − 2 x 2 − 2 x − 2 1 .
Finding the Constant Comparing this to f ( x ) = 2 x 5 + 5 x 4 + 2 x 3 − 4 x 2 − 4 x − 1 , we see that a = 2 . Therefore, f ( x ) = 2 ( x + 1 ) 3 ( x − 1 ) ( x + 2 1 ) . We can also write this as f ( x ) = ( x + 1 ) 3 ( x − 1 ) ( 2 x + 1 ) .
Final Answer The completely factored form of f ( x ) is f ( x ) = ( x + 1 ) 3 ( x − 1 ) ( 2 x + 1 ) .
Examples
Factoring polynomials is a fundamental concept in algebra with numerous real-world applications. For instance, in engineering, factoring can help analyze the stability of structures by finding the roots of characteristic equations. In economics, it can be used to model and predict market behavior by analyzing polynomial functions that represent supply and demand curves. Moreover, in computer graphics, factoring polynomials is essential for creating smooth curves and surfaces, which are used in various applications such as animation and computer-aided design.
The remaining zeros of the polynomial f ( x ) are x = 1 and x = − 2 1 . The completely factored form is f ( x ) = ( x + 1 ) 3 ( x − 1 ) ( 2 x + 1 ) .
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