Find all real solutions of the polynomial equation $x^4-6 x^3+30 x-25=0$.
A. $x=1,-6,-5$
B. $x=1,-25,10$
C. $x=1,5, \pm \sqrt{5}$
D. $x=1, \pm \sqrt{5}$
E. $x=1,25
Answer (1)
Verify that x = 1 is a root of the polynomial.
Perform polynomial division to factor out ( x − 1 ) .
Factor the resulting cubic polynomial by grouping.
Find all real roots: x = 1 , 5 , ± 5 .
x = 1 , 5 , ± 5
Explanation
Problem Analysis We are given the polynomial equation x 4 − 6 x 3 + 30 x − 25 = 0 and asked to find all real solutions.
Finding Potential Roots We can attempt to factor the polynomial or use the Rational Root Theorem to find possible rational roots. The possible rational roots are ± 1 , ± 5 , ± 25 .
Verifying x=1 as a Root Let's test x = 1 : 1 4 − 6 ( 1 ) 3 + 30 ( 1 ) − 25 = 1 − 6 + 30 − 25 = 0 . So, x = 1 is a root.
Polynomial Division Now we divide the polynomial by ( x − 1 ) to obtain a cubic polynomial. The polynomial division gives x 4 − 6 x 3 + 30 x − 25 = ( x − 1 ) ( x 3 − 5 x 2 − 5 x + 25 ) .
Factoring the Cubic Polynomial Next, we factor the cubic polynomial x 3 − 5 x 2 − 5 x + 25 . We can factor by grouping: x 2 ( x − 5 ) − 5 ( x − 5 ) = ( x 2 − 5 ) ( x − 5 ) . So, x 3 − 5 x 2 − 5 x + 25 = ( x − 5 ) ( x 2 − 5 ) .
Finding All Roots Therefore, the original polynomial is ( x − 1 ) ( x − 5 ) ( x 2 − 5 ) = 0 . The roots are x = 1 , x = 5 , and x 2 = 5 , which means x = ± 5 .
Real Solutions The real solutions are x = 1 , 5 , ± 5 .
Examples
Polynomial equations like the one we solved are used in various fields, such as physics, engineering, and economics, to model real-world phenomena. For example, in physics, they can describe the trajectory of a projectile or the behavior of electrical circuits. In economics, they can be used to model cost and revenue functions to optimize profits. Understanding how to solve polynomial equations is crucial for making accurate predictions and informed decisions in these fields.
