Introduction

To simplify a rational expression, we cancel out the largest common factor from the numerator and denominator. To do this, we sometimes need to factor the numerator and denominator before canceling.

For example, suppose that we wish to simplify \dfrac {4x - 12} {10x + 4}. To do this, we factor the numerator and denominator and cancel any factors that they have in common.

\eqalign{ \dfrac {4x - 12} {10x + 4} &=\dfrac {4(x - 3)} {2(5x + 2)}\\[5pt] &=\dfrac {2 \cdot 2 \cdot (x - 3)} {2(5x + 2)}\\[5pt] &=\dfrac {\cancel{2} \cdot 2(x - 3)} {\cancel{2}(5x + 2)}\\[5pt] &=\dfrac {2(x - 3)} {5x + 2} }

FLAG
Example: Simplifying Rational Expressions With Common Constant Factors

What is \dfrac {2x + 4} {2} reduced to lowest terms?

EXPLANATION

We factor the numerator and denominator and cancel any factors that they have in common.

\begin{eqnarray} \require{cancel} \eqalign{ \dfrac {2x + 4} {2} &= \dfrac {2(x + 2)} {2} \\[5pt] &= \dfrac {\cancel{2}(x + 2)} {\cancel{2} \cdot 1} \\[5pt] &= \dfrac {x + 2 } {1} \\[5pt] &= x + 2 } \end{eqnarray}

FLAG
Practice: Simplifying Rational Expressions With Common Constant Factors

What is $\dfrac{4z - 8x} {2z - 4y}$ reduced to its lowest terms?

a
 $\dfrac{2(z - 2x)} {z - 2y}$
b
 $4$
c
 $\dfrac{z - 2x} {z - 2y}$
d
 $\dfrac{z + 2x} {z + 2y}$
e
 $2$
Practice: Simplifying Rational Expressions With Common Constant Factors

$\dfrac {6b+6a} {6a}=$

a
 $b+a$
b
 $6b$
c
 $\dfrac {b+a} {a}$
d
 $b+1$
e
 $b$
Example: Simplifying Rational Expressions With Common Factors

Simplify the expression \dfrac {6y + 2} {3xy + x}.

EXPLANATION

We factor the numerator and denominator and cancel any factors that they have in common. \require{cancel} \eqalign{ \dfrac {6y + 2} {3xy + x} &= \dfrac {6y + 2} {x(3y + 1)} \\[5pt] &=\dfrac {2(3y + 1)} {x(3y + 1)} \\[5pt] &=\dfrac {2\cancel{(3y + 1)} } {x \cancel{(3y + 1)} } \\[5pt] &=\dfrac {2 } {x} }

FLAG
Practice: Simplifying Rational Expressions With Common Factors

$\dfrac{4xy - x} {4y - 1}=$

a
 $\dfrac{x}{y}$
b
 Not reducible
c
 $\dfrac{1}{y}$
d
 $xy$
e
 $x$
Practice: Simplifying Rational Expressions With Common Factors

$\dfrac{ab - b} {3a - 3}=$

a
 $\dfrac{b}{a}$
b
 Not reducible
c
 $\dfrac{b}{3}$
d
 $\dfrac{a}{b}$
e
 $\dfrac{3}{b}$
Example: Identifying Irreducible Rational Expressions

What is \dfrac{x+2}{2x} reduced to lowest terms?

EXPLANATION

The numerator and denominator have no common factors, so the expression can not be reduced. The expression \dfrac{x+2}{2x} is already in lowest terms.

FLAG
Practice: Identifying Irreducible Rational Expressions

$\dfrac {7x + 1} {7x}=$

a
 $1$
b
 $7x + 1$
c
 $\dfrac {7x + 1} {7x}$
d
 $7x$
e
 $7$
Practice: Identifying Irreducible Rational Expressions

$\dfrac {3a+4} {2}=$

a
 $2$
b
 $3a+2$
c
 $3a+4$
d
 $\dfrac{3a}{2}$
e
 $\dfrac {3a+4} {2}$
Example: Simplifying Rational Expressions With Longer Numerators and Denominators

Simplify \dfrac{5c + 10p - 20}{5p+15}.

EXPLANATION

We factor the numerator and denominator and cancel any factors that they have in common. \require{cancel} \begin{align*} \dfrac{5c + 10p - 20}{5p+15} &= \dfrac{5(c+2p-4)}{5(p+3)} \\[5pt] &= \dfrac{\cancel{5}(c+2p-4)}{\cancel{5}(p+3)} \\[5pt] &= \dfrac{c+2p-4}{p+3} \end{align*}

FLAG
Practice: Simplifying Rational Expressions With Longer Numerators and Denominators

What is $\dfrac{4ab-8+10x}{10x+8-2ab}$ reduced to its lowest terms?

a
 $-2$
b
 $\dfrac{2ab-4+5x}{5x+4-ab}$
c
 $2$
d
 $\dfrac{ab-2+5x}{5x+2-ab}$
e
 $\dfrac{ab-4+5x}{5x+4-ab}$
Practice: Simplifying Rational Expressions With Longer Numerators and Denominators

What is $\dfrac{8a-4b+12}{20b+ 8a}$ reduced to its lowest terms?

a
 $\dfrac{-4b+12}{20b}$
b
 $\dfrac{-b+5}{4b}$
c
 $\dfrac{2a-b}{5b+2a}$
d
 $\dfrac{2a+b-3}{5b-2a}$
e
 $\dfrac{2a-b+3}{5b+2a}$
Flag Content
Did you notice an error, or do you simply believe that something could be improved? Please explain below.
SUBMIT
CANCEL