REDUCING RATIONAL EXPRESSIONS TO LOWEST TERMS

To reduce rational expressions to lowest terms, we will follow the steps given below.

  • Factorize the numerator and denominator as much as possible.
  • Cancel the common factors in the numerator and denominator.
  • Write down the left over terms.

Factorize and simplify:

Problem 1 :

3a+123

Solution:

=3a+123= 3(a + 4)3= a+4

Problem 2 :

5x-15x-3

Solution:

=5x-15x-3=5(x-3)x-3=5

Problem 3 :

a+5a2+7a+10

Solution:

=a+5a2+7a+10=a+5(a+5) (a+2)=1a+2

Problem 4 :

m2-mm2-1

Solution:

=m2-mm2-1=m(m-1)(m+1) (m-1)= mm+1

Problem 5 :

n2-n-6n2+5n+6

Solution:

=n2-n-6n2+5n+6=(n+2) (n-3)(n+2) (n+3)=n-3n+3

Problem 6 :

2x2-x-34x2-9

Solution:

=2x2-x-34x2-9=2x2-3x+2x-3(2x)2-32=x(2x-3)+1(2x-3)(2x+3)(2x-3)=(2x-3)(x+1)(2x-3)(2x+3)=x+12x+3

Simplify:

Problem 7:

3x+152×4xx+5

Solution:

=3x+152×4xx+5=3(x+5)2×4xx+5=6x

Problem 8 :

a2-9a-3×a+1a+3

Solution:

a2-9a-3×a+1a+3

Factor the expression using a2 - b2 = (a + b) (a - b)

=(a+3)(a-3)a-3×a+1a+3=a+1

Problem 9:

3x+610x÷x+25x

Solution:

3x+610x÷x+25x

Divide a fraction by multiplying its reciprocal,

=3x+610x×5xx+2= 3(x+2)10x×5xx+2=32

Problem 10:

m2-25m2-5m÷m+55m

Solution:

m2-25m2-5m÷m+55m

Factor the expression using a2 - b2 = (a + b) (a - b)

=(m+5)(m-5)m(m-5)×5mm+5=5

Problem 11 :

a2+7a+12a2+5a+4×a2+6a+5a2+12a+35

Solution:

a2+7a+12a2+5a+4×a2+6a+5a2+12a+35

Factor the expression,

=(a+3) (a+4)(a+1) (a+4)×(a+1) (a+5)(a+5) (a+7)=a+3a+7

Problem 12 :

n2-3n-43n2-48÷n3-nn2+4n

Solution:

=n2-3n-43n2-48÷n3-nn2+4n

Factor the expression,

=n2-3n-43n2-48×n2+4nn3-n=(n+1) (n-4)3n2-16×n(n+4)nn2-1=(n+1) (n-4)3n2-16×n(n+4)nn2-1=(n+1) (n-4)3(n+4) (n-4)×(n+4)(n+1) (n-1)=13(n-1)

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