Question

1. Simplify the following as a quotient of two polynomials in the simplest form.

Answer 1

SOLUTION IS IN THE ATTACHMENT

RATIONAL EXPRESSIONS :
A rational number is defined as a quotient a/b, of two integers a and b and b ≠0.A rational expression is a quotient p(x) /q(x) of two Polynomials p(x) and q(x) ,where q(x) ≠0.
•When the numerator and denominator of a rational expression do not have any common factor except 1, the rational expression is said to be expressed in the lowest terms.
•To reduce a rational expression to the lowest terms, factorise the numerator and the denominator and cancel the factors which are common to both.

HOPE THIS WILL HELP YOU….

Answer 2

Solution :

i ) x² + 3x + 2

Splitting the middle term ,

= x² + x + 2x + 2

= x( x + 1 ) + 2 ( x + 1 )

= ( x + 1 )( x + 2 ) -----( 1 )
_____________________
ii ) x² + 5x + 6

= x² + 2x + 3x + 6

= x( x + 2 ) + 3( x + 2 )

= ( x + 2 )( x + 3 ) ----( 2 )
_____________________
iii ) x² + 4x + 3

= x² + x + 3x + 3

= x( x + 1 ) + 3( x + 1 )

= ( x + 1 )( x + 3 ) ------- ( 3 )
______________________
Now , take first two terms

1/(x²+3x+2) + 1/(x²+5x+6)

[ from ( 1 ) and ( 2 ) ]

= 1/[(x+1)(x+2)] + 1/[(x+2)(x+3)]

= 1/(x+2)[1/(x+1) + 1/(x+3)]

= 1/(x+2)[(x+3+x+1)/(x+1)(x+3)]

= [( 2x+4 )/(x+1)(x+2)(x+3)]

= [2(x+2)]/[(x+1)(x+2)(x+3)]

= 2/[( x+1)(x+3) ]------( 4 )
_________________________
Now , take 3 terms

[1/(x²+3x+2)+1/(x²+5x+6)] -2/(x²+4x+3)

[ from ( 4 ) ]

= 2/(x+1)(x+3) - 2/(x+1)(x+3)

[ from (4)and (3)]

= 0

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