Find the rational zeros of the polynomial function and write in factored form.
[tex]f(x)=2 x^3-3 x^2-5 x+6[/tex]
A. [tex]f(x)=(x+1)(x+2)(2 x-3)[/tex]
B. [tex]f(x)=(x-1)(x+2)(2 x-3)[/tex]
C. [tex]f(x)=(x-1)(x+1) 2(x-3)[/tex]
D. [tex]f(x)=(x-1)(x-2)(2 x+3)[/tex]
Answer (2)
The rational zeros of the polynomial f ( x ) = 2 x 3 − 3 x 2 − 5 x + 6 are found to be 1, -1, and 2, leading to the factors (x - 1), (x + 1), and (x - 2). The polynomial in factored form is expressed as f ( x ) = ( x − 1 ) ( x + 2 ) ( 2 x − 3 ) , hence the correct answer is option B.
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Find the roots of the polynomial: x = − 2 3 , 1 , 2 .
Determine the factors corresponding to the roots: ( 2 x + 3 ) , ( x − 1 ) , ( x − 2 ) .
Write the polynomial in factored form: f ( x ) = ( x − 1 ) ( x − 2 ) ( 2 x + 3 ) .
The correct factored form is f ( x ) = ( x − 1 ) ( x − 2 ) ( 2 x + 3 ) .
Explanation
Problem Analysis We are given the polynomial function f ( x ) = 2 x 3 − 3 x 2 − 5 x + 6 and asked to find its rational zeros and factored form. We are also given four possible factored forms to choose from.
Finding the Roots First, let's find the rational roots of the polynomial. From the previous calculation, we know that the roots are -1.5, 1, and 2.
Expressing Roots as Fractions Now, let's express the roots as fractions: -1.5 = -3/2, 1 = 1, and 2 = 2.
Finding the Factors The roots correspond to the factors of the polynomial. If x = a is a root, then ( x − a ) is a factor. So, we have the factors ( x − 1 ) and ( x − 2 ) . For the root x = − 2 3 , we have ( x − ( − 2 3 )) = ( x + 2 3 ) . To get rid of the fraction, we can multiply this factor by 2 to get ( 2 x + 3 ) .
Factored Form Therefore, the factored form of the polynomial is ( x − 1 ) ( x − 2 ) ( 2 x + 3 ) .
Final Answer Comparing this with the given options, we see that the correct factored form is f ( x ) = ( x − 1 ) ( x − 2 ) ( 2 x + 3 ) .
Examples
Factoring polynomials is useful in many real-world applications, such as designing structures, modeling population growth, or optimizing business processes. For example, if you are designing a bridge, you might use polynomial functions to model the load on the bridge and ensure that it can withstand the forces acting on it. Factoring the polynomial can help you find the critical points where the load is highest, allowing you to design a stronger and safer bridge. Similarly, in business, you might use polynomial functions to model the cost of production and find the optimal level of production that minimizes costs and maximizes profits.
