If [tex]u(x)=x^5-x^4+x^2[/tex] and [tex]v(x)=-x^2[/tex], which expression is equivalent to [tex](\frac{u}{v})(x)[/tex]?
A. [tex]x^3-x^2[/tex]
B. [tex]-x^3+x^2[/tex]
C. [tex]-x^3+x^2-1[/tex]
D. [tex]x^3-x^2+1[/tex]
Answer (1)
Divide u ( x ) by v ( x ) : v ( x ) u ( x ) = − x 2 x 5 − x 4 + x 2 .
Divide each term in the numerator by − x 2 .
Simplify each term: − x 2 x 5 = − x 3 , − x 2 − x 4 = x 2 , and − x 2 x 2 = − 1 .
Combine the simplified terms to get the final expression: − x 3 + x 2 − 1 .
Explanation
Understanding the Problem We are given two functions, u ( x ) = x 5 − x 4 + x 2 and v ( x ) = − x 2 . We need to find an expression equivalent to ( v u ) ( x ) , which means we need to divide u ( x ) by v ( x ) .
Setting up the Division We have ( v u ) ( x ) = v ( x ) u ( x ) = − x 2 x 5 − x 4 + x 2 . To simplify this expression, we can divide each term in the numerator by − x 2 .
Performing the Division Now, let's divide each term:
− x 2 x 5 = − x 5 − 2 = − x 3
− x 2 − x 4 = x 4 − 2 = x 2
− x 2 x 2 = − 1
Combining the Terms Combining these results, we get − x 3 + x 2 − 1 .
Final Answer Therefore, ( v u ) ( x ) = − x 3 + x 2 − 1 .
Examples
Understanding how to divide polynomials is useful in many areas of mathematics and engineering. For example, when designing circuits, engineers often use polynomial functions to model the behavior of the circuit. Simplifying these functions by division can help in analyzing and optimizing the circuit's performance. Also, in computer graphics, polynomial division can be used to manipulate curves and surfaces.
