Find the LCM and GCD for the following and verify that f ( x ) × g ( x ) = L C M × G C D x 2 y + x y 2 , x 2 + x y

Let the given expressions be, f ( x ) = x 2 y + x y 2 g ( x ) = x 2 + x y Factorizing the given expressions, f ( x ) = x 2 y + x y 2 = x y ( x + y ) g ( x ) = x 2 + x y = x ( x + y ) L C M f x , g x = x y ( x + y ) , and G C D f x , g x = x ( x + y ) Now, verifying the equation: f ( x ) × g ( x ) = L C M × G C D LHS = f ( x ) × g ( x ) = x y x + y x x + y = x y × x x + y x + y = x 2 y x + y 2 RHS L C M × G C D = x y ( x + y ) × x ( x + y ) = x y × x x + y x + y = x 2 y x + y 2 LHS = RHS Hence, the statement has been verified.

Are x 2 - 1 and tan x = sin x cos x rational expressions?

An expression is called a rational expression if it can be written in the form p ( x ) q ( x ) where p ( x ) and q ( x ) are polynomials and q ( x ) ≠ 0 . A rational expression is the ratio of two polynomials. x 2 - 1 can be written as x 2 - 1 1 Here x 2 - 1 and 1 both are polynomials. Therefore x 2 - 1 is a rational expression. Whereas, tan x , sin x and cos x are trigonometric functions, they are not polynomials. Therefore, tan x , sin x and cos x are not rational expressions.

Reduce the rational expressions to its lowest form. x 2 - 11 x + 18 x 2 - 4 x + 4

Given expression is x 2 - 11 x + 18 x 2 - 4 x + 4 . A rational expression p x q x is said to be in the lowest form if G C D p x , q x = 1 . Here, p ( x ) = x 2 - 11 x + 18 and q ( x ) = x 2 - 4 x + 4 . Reducing the given expression to the lowest form, x 2 - 11 x + 18 x 2 - 4 x + 4 = ( x - 2 ) ( x - 9 ) ( x - 2 ) ( x - 2 ) = x - 9 x - 2 Hence, the reduced form the given expression is x - 9 x - 2 .

Reduce the following rational expression to its least form. 9 x 2 + 81 x x 3 + 8 x 2 - 9 x

Given expression is 9 x 2 + 81 x x 3 + 8 x 2 - 9 x . A rational expression p x q x is said to be in the lowest form if G C D p x , q x = 1 . Here, p ( x ) = 9 x 2 + 81 x and q ( x ) = x 3 + 8 x 2 - 9 x . Reducing the given expression to the lowest form, 9 x 2 + 81 x x 3 + 8 x 2 - 9 x = 9 x ( x + 9 ) x x 2 + 8 x - 9 = 9 ( x + 9 ) ( x + 9 ) ( x - 1 ) = 9 x - 1 Hence, the reduced form is 9 x - 1 .

Reduce the following rational expression to its lowest form. p 2 - 3 p - 40 2 p 3 - 24 p 2 + 64 p

Given expression is p 2 - 3 p - 40 2 p 3 - 24 p 2 + 64 p . A rational expression p x q x is said to be in the lowest form if G C D p x , q x = 1 . Reducing the given to the lowest form, p 2 - 3 p - 40 2 p 3 - 24 p 2 + 64 p = ( p - 8 ) ( p + 5 ) 2 p p 2 - 12 p + 32 = ( p - 8 ) ( p + 5 ) 2 p ( p - 4 ) ( p - 8 ) = p + 5 2 p ( p - 4 ) Hence, the reduced form is p + 5 2 p ( p - 4 ) .