One root of [tex]f(x)=x^3+10 x^2-25 x-250[/tex] is [tex]x=-10[/tex]. What are all the roots of the function? Use the Remainder Theorem.
A. [tex]x=-25[/tex] or [tex]x=10[/tex]
B. [tex]x=-25, x=1[/tex], or [tex]x=10[/tex]
C. [tex]x=-10[/tex] or [tex]x=5[/tex]
D. [tex]x=-10, x=-5[/tex], or [tex]x=5[/tex]
Answer (1)
Factor the polynomial using the given root: f ( x ) = ( x + 10 ) ( x 2 − 25 ) .
Factor the quadratic term: x 2 − 25 = ( x − 5 ) ( x + 5 ) .
Find the roots by setting each factor to zero: x + 10 = 0 , x − 5 = 0 , x + 5 = 0 .
The roots are: x = − 10 , x = − 5 , x = 5 .
Explanation
Understanding the Problem We are given the function f ( x ) = x 3 + 10 x 2 − 25 x − 250 and told that x = − 10 is one of its roots. Our goal is to find all the roots of this function. Since we know one root, we can use the Remainder Theorem to factor the polynomial.
Polynomial Division Since x = − 10 is a root, we know that ( x + 10 ) is a factor of f ( x ) . We can perform polynomial division to find the other factor. Dividing x 3 + 10 x 2 − 25 x − 250 by ( x + 10 ) , we get: ( x 3 + 10 x 2 − 25 x − 250 ) \tdiv ( x + 10 ) = x 2 − 25
Factoring the Quadratic Now we have factored the polynomial as f ( x ) = ( x + 10 ) ( x 2 − 25 ) . To find the remaining roots, we need to solve x 2 − 25 = 0 . This is a difference of squares, so we can factor it as ( x − 5 ) ( x + 5 ) = 0 .
Finding the Roots Setting each factor to zero, we get x − 5 = 0 or x + 5 = 0 . Solving for x , we find x = 5 or x = − 5 . Therefore, the roots of the quadratic factor are x = 5 and x = − 5 .
Final Answer Thus, the roots of the function f ( x ) = x 3 + 10 x 2 − 25 x − 250 are x = − 10 , x = 5 , and x = − 5 .
Examples
Polynomials are used to model curves and shapes in various fields, such as engineering, physics, and computer graphics. For example, engineers use polynomials to design bridges and buildings, ensuring stability and optimal performance. In computer graphics, polynomials are used to create smooth curves and surfaces for 3D models and animations. Understanding how to find the roots of a polynomial helps in determining key points and characteristics of these models.
