Find a rational zero of the polynomial function and use it to find all the zeros of the function. [tex]f(x)=x^4-4 x^3+13 x^2+64 x-464[/tex]
Answer (2)
Apply the Rational Root Theorem to identify potential rational roots.
Test the potential roots to find a rational zero, which is x = 4 .
Use synthetic division to reduce the polynomial to x 3 + 13 x + 116 .
Find another rational root for the cubic polynomial, which is x = − 4 .
Use synthetic division again to reduce the polynomial to x 2 − 4 x + 29 .
Solve the quadratic equation to find the remaining roots 2 ± 5 i .
The zeros of the polynomial are 4 , − 4 , 2 + 5 i , 2 − 5 i .
Explanation
Problem Analysis We are given the polynomial function f ( x ) = x 4 − 4 x 3 + 13 x 2 + 64 x − 464 and asked to find a rational zero and then all zeros of the function.
Rational Root Theorem We will use the Rational Root Theorem to find potential rational zeros. The possible rational roots are of the form ± factors of 1 factors of 464 . The factors of 464 are 1, 2, 4, 8, 16, 29, 58, 116, 232, and 464.
Testing Potential Roots We test the potential rational roots by substituting them into f ( x ) . Let's start with x = 4 :
f ( 4 ) = ( 4 ) 4 − 4 ( 4 ) 3 + 13 ( 4 ) 2 + 64 ( 4 ) − 464 = 256 − 256 + 208 + 256 − 464 = 0 Since f ( 4 ) = 0 , x = 4 is a rational root.
Synthetic Division Now we use synthetic division to divide f ( x ) by ( x − 4 ) .
Coefficients of f ( x ) : 1, -4, 13, 64, -464
Performing synthetic division with x = 4 :
4 | 1 -4 13 64 -464 | 4 0 52 464 ------------------------ 1 0 13 116 0
The quotient is x 3 + 0 x 2 + 13 x + 116 = x 3 + 13 x + 116 .
Finding Another Root Now we need to find the zeros of the cubic polynomial x 3 + 13 x + 116 = 0 . We can try to find another rational root. The factors of 116 are 1, 2, 4, 29, 58, and 116. Let's test x = − 4 :
( − 4 ) 3 + 13 ( − 4 ) + 116 = − 64 − 52 + 116 = 0 . So, x = − 4 is a root.
Synthetic Division Again Now we use synthetic division again to divide x 3 + 13 x + 116 by ( x + 4 ) .
-4 | 1 0 13 116 | -4 16 -116 ------------------ 1 -4 29 0
The quotient is x 2 − 4 x + 29 .
Quadratic Formula Now we solve the quadratic equation x 2 − 4 x + 29 = 0 using the quadratic formula: x = 2 a − b ± b 2 − 4 a c = 2 ( 1 ) 4 ± ( − 4 ) 2 − 4 ( 1 ) ( 29 ) = 2 4 ± 16 − 116 = 2 4 ± − 100 = 2 4 ± 10 i = 2 ± 5 i So the remaining roots are 2 + 5 i and 2 − 5 i .
Final Answer The zeros of the polynomial f ( x ) = x 4 − 4 x 3 + 13 x 2 + 64 x − 464 are 4 , − 4 , 2 + 5 i , and 2 − 5 i .
Examples
Polynomial functions are used in various fields such as physics, engineering, and economics. For example, in physics, projectile motion can be modeled using a quadratic polynomial function. In engineering, polynomial functions can be used to approximate complex curves and surfaces. In economics, cost and revenue functions can be modeled using polynomial functions to analyze business performance. Finding the roots of these polynomial functions helps in determining key values such as maximum height, optimal design parameters, or break-even points.
The rational zeros of the polynomial function f ( x ) = x 4 − 4 x 3 + 13 x 2 + 64 x − 464 are found to be 4 and − 4 . The other zeros are complex: 2 + 5 i and 2 − 5 i . Hence, the complete set of zeros includes 4 , − 4 , 2 + 5 i , 2 − 5 i .
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