Find the rational zeros of the polynomial function and write in factored form. [tex]f(x)=-x^3+5 x^2-8 x+4[/tex]
Answer (2)
The rational zeros of the polynomial function f ( x ) = − x 3 + 5 x 2 − 8 x + 4 are x = 1 and x = 2 (with multiplicity 2). The factored form of the polynomial is f ( x ) = − ( x − 1 ) ( x − 2 ) 2 .
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Test each of the provided factored forms by expanding them.
Identify the correct factored form: f ( x ) = − ( x − 1 ) ( x − 2 ) 2 .
Find the rational zeros by setting each factor to zero: x = 1 and x = 2 .
State the rational zeros and the factored form: f ( x ) = − ( x − 1 ) ( x − 2 ) 2
Explanation
Problem Analysis We are given the polynomial function f ( x ) = − x 3 + 5 x 2 − 8 x + 4 and asked to find its rational zeros and factored form from the given options. We will test each of the provided factored forms by expanding them and comparing the result to the original polynomial.
Testing Option 1 Let's test the first option: f ( x ) = − ( x − 1 ) ( x − 4 ) 2 . Expanding this gives
− ( x − 1 ) ( x − 4 ) 2 = − ( x − 1 ) ( x 2 − 8 x + 16 ) = − ( x 3 − 8 x 2 + 16 x − x 2 + 8 x − 16 ) = − ( x 3 − 9 x 2 + 24 x − 16 ) = − x 3 + 9 x 2 − 24 x + 16 . This does not match the original polynomial, so it is incorrect.
Testing Option 2 Now let's test the second option: f ( x ) = ( x − 1 ) ( 2 − x ) 2 . Expanding this gives
( x − 1 ) ( 2 − x ) 2 = ( x − 1 ) ( 4 − 4 x + x 2 ) = ( 4 x − 4 x 2 + x 3 − 4 + 4 x − x 2 ) = x 3 − 5 x 2 + 8 x − 4 . This also does not match the original polynomial, so it is incorrect.
Testing Option 3 Next, let's test the third option: f ( x ) = − ( x − 1 ) ( x − 2 ) 2 . Expanding this gives
− ( x − 1 ) ( x − 2 ) 2 = − ( x − 1 ) ( x 2 − 4 x + 4 ) = − ( x 3 − 4 x 2 + 4 x − x 2 + 4 x − 4 ) = − ( x 3 − 5 x 2 + 8 x − 4 ) = − x 3 + 5 x 2 − 8 x + 4 . This matches the original polynomial, so it is the correct factored form.
Testing Option 4 Finally, let's test the fourth option: f ( x ) = − ( x − 1 ) ( x + 2 ) 2 . Expanding this gives
− ( x − 1 ) ( x + 2 ) 2 = − ( x − 1 ) ( x 2 + 4 x + 4 ) = − ( x 3 + 4 x 2 + 4 x − x 2 − 4 x − 4 ) = − ( x 3 + 3 x 2 − 4 ) = − x 3 − 3 x 2 + 4 . This does not match the original polynomial, so it is incorrect.
Finding Rational Zeros The correct factored form is f ( x ) = − ( x − 1 ) ( x − 2 ) 2 . To find the rational zeros, we set each factor to zero:
x − 1 = 0 ⟹ x = 1
x − 2 = 0 ⟹ x = 2 . Since the factor ( x − 2 ) is squared, the zero x = 2 has a multiplicity of 2.
Therefore, the rational zeros are x = 1 and x = 2 .
Final Answer The rational zeros of the polynomial function f ( x ) = − x 3 + 5 x 2 − 8 x + 4 are 1 and 2 , and the factored form is f ( x ) = − ( x − 1 ) ( x − 2 ) 2 .
Examples
Understanding polynomial functions and their zeros is crucial in many areas of science and engineering. For example, in physics, the trajectory of a projectile can be modeled by a polynomial function, and finding the zeros of this function can tell us when the projectile will hit the ground. In engineering, polynomial functions are used to model the behavior of circuits and systems, and finding the zeros can help us understand the stability and performance of these systems. Factoring polynomials and finding their zeros allows engineers to design stable and efficient systems.
