SOLVING PROBLEMS INVOLVING RATIONAL ALGEBRAIC EXPRESSIONS

Add/Subtract the following rational expressions:

Problem 1 :

xx+2+3x-4

Solution:

=xx+2+3x-4=x(x-4)(x+2)(x-4)+3(x+2)(x+2)(x-4)=x2-4x+(3x+6)(x+2)(x-4)=x2-x+6x2-2x-8

Problem 2 :

x2+1(x-1)(x-2)-x3x-3

Solution:

=x2+1(x-1)(x-2)-x3x-3=x2+1(x-3)(x-1)(x-2)(x-3)-x3(x-1)(x-2)(x-1)(x-2)(x-3)=x3-3x2+x-3-x5-3x4+2x3(x-1)(x-2)(x-3)=-x5+3x4-x3-3x2+x-3(x-1)x2-5x+6=-x5+3x4-x3-3x2+x-3x3-6x2+11x-6

Simplify the following rational expressions (if possible):

Problem 3 :

x2+x-2x2-1

Solution:

=x2+x-2x2-1=(x+2)(x-1)(x+1)(x-1)=(x+2)(x+1)

Problem 4 :

x2+5x+6x2-3x+2

Solution:

=x2+5x+6x2-3x+2=(x+3)(x+2)(x-1)(x-2)

Problem 5 :

xx+2+3x+1x-1

Solution:

=xx+2+3x+1x-1=xx+2+3x+1x-1×(x+2)(x-1)(x+2)(x-1)=x(x-1)+3(x+2)(x-1)(x+1)(x+2)=x2-x+3x2+3x-6x2+3x+2=4x2+2x-6x2+3x-2

Perform the operation indicated and simplify each expression. Identify any domain restrictions.

Problem 6 :

x2+xx+24-x2x2+3x

Solution:

=x2+xx+24-x2x2+3x=x(x+1)x+24-x2x(x+3)=x(x+1)x+2×(2+x)(2-x)x(x+3)=(x+1)×(2-x)(x+3)=(x+1)(2-x)(x+3)

Problem 7 :

x2-6x+9x2-9÷x-9x+3

Solution:

=x2-6x+9x2-9÷x-9x+3=(x-3)2x2-9÷x-9x+3=(x-3)2(x+3)(x-3)÷x-9x+3=x-3x+3÷x-9x+3=x-3x+3×x+3x-9=(x-3)(x-9

Problem 8 :

4x2-9-2x2-3x

Solution:

=4x2-9-2x2-3x=4(x+3)(x-3)-2x2-3x=4(x+3)(x-3)-2x(x-3)=4x-2(x+3)x(x+3)(x-3)=4x-2x-6x(x+3)(x-3)=2x-6x(x+3)(x-3)=2(x-3)x(x+3)(x-3)=2x(x+3)

Problem 9 :

x+2-4xx+2

Solution:

=x+2-4xx+2=x(x+2)+2(x+2)-4xx+2=x2+2x+2x+4-4xx+2=x2+4x+2

Problem 10 :

1x+1-1x

Solution:

=1x+1-1x=1-(x+1)x+1x=1-x-1x+1x=-xx+1x=-xx+1×1x=-1x+1

Problem 11 :

x-1x1-x2x

Solution:

=x-1x1-x2x=x2-1x1-x2x=x2-1x×x1-x2=x2-1-x2-1=-1

Problem 12 :

1x-2+12x

Solution:

=1x-2+12x=2+x-22(x-2)x=x2(x-2)x=x2(x-2)×1x=12(x-2)

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