Find all complex zeros of the given polynomial function, and write the polynomial in completely factored form.
[tex]f(x)=3 x^3-7 x^2-43 x+15[/tex]
Find the complex zeros of [tex]f[/tex]. Repeat any zeros if their multiplicity is greater than 1.
[tex]x=-3, \frac{1}{3}, 5[/tex]
(Simplify your answer. Use a comma to separate answers as needed. Express complex numbers in terms of [tex]i[/tex]. Use integers or fractions for any numbers in the expression.) Use the complex zeros to factor f.
[tex]f(x)=[/tex]
(Simplify your answer. Type your answer in factored form. Express complex numbers in terms of [tex]i[/tex]. Use integers or fractions for any numbers in the expression.)
Answer (1)
The polynomial function is f ( x ) = 3 x 3 − 7 x 2 − 43 x + 15 and its zeros are x = − 3 , 3 1 , 5 .
Write the polynomial in factored form using the zeros: f ( x ) = a ( x − r 1 ) ( x − r 2 ) ( x − r 3 ) .
Substitute the given zeros: f ( x ) = a ( x + 3 ) ( x − 3 1 ) ( x − 5 ) .
Determine the leading coefficient a = 3 and substitute it into the factored form: f ( x ) = ( x + 3 ) ( 3 x − 1 ) ( x − 5 ) .
The completely factored form of the polynomial is ( x + 3 ) ( 3 x − 1 ) ( x − 5 ) .
Explanation
Understanding the Problem We are given the polynomial function f ( x ) = 3 x 3 − 7 x 2 − 43 x + 15 and its zeros x = − 3 , 3 1 , 5 . Our goal is to write the polynomial in completely factored form using these zeros.
Factored Form of Polynomial A polynomial can be written in factored form as f ( x ) = a ( x − r 1 ) ( x − r 2 ) ( x − r 3 ) , where r 1 , r 2 , r 3 are the zeros of the polynomial and a is the leading coefficient. In our case, the zeros are − 3 , 3 1 , 5 .
Substituting the Zeros Substituting the zeros into the factored form, we get f ( x ) = a ( x − ( − 3 )) ( x − 3 1 ) ( x − 5 ) , which simplifies to f ( x ) = a ( x + 3 ) ( x − 3 1 ) ( x − 5 ) .
Finding the Leading Coefficient The leading coefficient of the given polynomial f ( x ) = 3 x 3 − 7 x 2 − 43 x + 15 is 3. Therefore, a = 3 .
Final Factored Form Substituting a = 3 into the factored form, we have f ( x ) = 3 ( x + 3 ) ( x − 3 1 ) ( x − 5 ) . We can also write this as f ( x ) = ( x + 3 ) ( 3 x − 1 ) ( x − 5 ) .
Examples
Factoring polynomials is a fundamental concept in algebra and has numerous real-world applications. For instance, engineers use polynomial factorization to analyze the stability of structures and systems. Imagine designing a bridge; engineers use polynomials to model the forces acting on the bridge. By factoring these polynomials, they can identify critical points where the structure might be weak or unstable, ensuring the bridge's safety and durability. Similarly, in control systems, factoring polynomials helps in determining the stability of a system, ensuring it operates reliably under various conditions. This technique is also crucial in cryptography, where factoring large numbers (which can be represented as polynomials) is essential for secure data transmission.
