SOLVING PROBLEMS INVOLVING RATIONAL ALGEBRAIC EXPRESSIONS
Add/Subtract the following rational expressions:
Problem 1 :
Solution:
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+2⋅4-x2x2+3x
Solution:
=x2+xx+2⋅4-x2x2+3x=x(x+1)x+2⋅4-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)