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 (2)
The equivalent expression for ( v u ) ( x ) is − x 3 + x 2 − 1 , which corresponds to option C. By dividing u ( x ) by v ( x ) and simplifying, we arrive at this result. Hence, the correct answer is option C.
;
Divide u ( x ) by v ( x ) : v ( x ) u ( x ) = − x 2 x 5 − x 4 + x 2 .
Simplify each term: − x 2 x 5 = − x 3 , − x 2 − x 4 = x 2 , − x 2 x 2 = − 1 .
Combine the terms: − x 3 + x 2 − 1 .
The equivalent expression is − 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 the expression equivalent to ( v u ) ( x ) , which means we need to divide u ( x ) by v ( x ) .
Dividing the Functions To find ( v u ) ( x ) , we divide u ( x ) by v ( x ) : v ( x ) u ( x ) = − x 2 x 5 − x 4 + x 2 .
Simplifying the Expression Now, we simplify the expression by dividing each term in the numerator by − x 2 : − x 2 x 5 − − x 2 x 4 + − x 2 x 2 = − x 3 + x 2 − 1 .
Finding the Equivalent Expression So, ( v u ) ( x ) = − x 3 + x 2 − 1 . Comparing this with the given options, we find that it matches the third option.
Examples
Understanding how to divide polynomials is useful in many areas of mathematics and engineering. For example, when designing a bridge, engineers use polynomial functions to model the load and stress on different parts of the structure. Dividing these polynomials can help them determine the optimal shape and size of the bridge components to ensure stability and safety. Similarly, in computer graphics, polynomial division is used to create smooth curves and surfaces for realistic 3D models.
