5. Rational Functions & Expressions

Lesson

When we looked at how to find the least common denominator we learned how to add and subtract fractions. To find the least common denominator (LCD), we need to find the least common multiple between the denominators.

For example, let's say we want to find the LCD between $\frac{1}{4}$14 and $\frac{3}{10}$310. We could multiply the denominators together to get a denominator of $40$40. However, this is not the *least* common denominator.

We can look at factors that the terms already have in common, then multiply the terms

$\frac{1}{4}=\frac{1}{2\times2}$14=12×2 and $\frac{3}{10}=\frac{3}{2\times5}$310=32×5

So $2$2 is a common factor. Now we can take this common factor and multiply it by any uncommon factors from the terms to find our LCD.

$2\times2\times5=20$2×2×5=20, so $20$20 is our least common denominator.

We can use this same process to simplify algebraic fractions. Let's look at an example:

Find the LCD for $\frac{12}{4x}$124`x` and $\frac{6}{20x^3}$620`x`3.

1. Find the LCM between the coefficients.

Our coefficients are $4$4 and $20$20. $4\times5=20$4×5=20 and $20\times1=20$20×1=20, so the LCM is $20$20 of the coefficients.

2. Find the LCM of the variables.

Our variables are $x$`x` and $x^3$`x`3.

$x\times x^2=x^3$`x`×`x`2=`x`3 and $x^3\times1=x^3$`x`3×1=`x`3, so the LCM is $x^3$`x`3.

3. Multiply the LCMs together.

The least common multiple is $20x^3$20`x`3.

Handy Hint

It may be helpful to factor algebraic terms first before you look to find the LCD,

so make sure you're familiar with how to factor different kinds of algebraic expressions.

Find the least common denominator for this pair of algebraic fractions:

$\frac{1}{18h}$118`h` and $\frac{1}{9h}$19`h`

In an upcoming election, it is anticipated that the number of men who vote, $x$`x`, will be greater than the number of women who vote, $y$`y`.

Each expression below represents the expected number of people who will NOT vote in two different counties.

Which county expects more people to not vote?

$\frac{x-y}{2}$

`x`−`y`2A$x-\frac{y}{2}$

`x`−`y`2B$\frac{x-y}{2}$

`x`−`y`2A$x-\frac{y}{2}$

`x`−`y`2B

Find the least common denominator of this pair of fractions:

$\frac{1}{q\left(q+5\right)}$1`q`(`q`+5) and $\frac{1}{q\left(q-2\right)}$1`q`(`q`−2)

The most important things to remember when adding and subtracting fractions (of any kind) are

- we need
**like denominators** - we need to keep our fractions
**equivalent**

Now we are going to build on this knowledge and look at how to add and subtract algebraic fractions.

$\frac{4m}{5}-\frac{2}{5}$4`m`5−25

**Think**: The first thing we need to do is check that the denominators are the same. In this case, both denominators are $5$5, so then we move on to the next step.

**Do**: Because our denominators are the same, we can write the numerators together as a single expression over the common denominator.

$\frac{4m-2}{5}$4`m`−25

**Reflect**: Because we cannot simplify the numerator $4m-2$4`m`−2 any further, this means this is a simplified as we can get.

Let's look at a very similar example, but where the denominators are not initially the same.

$\frac{2y}{3}+\frac{5}{6}$2`y`3+56

**Think**: We cannot add or subtract fractions unless the denominators are common. In this case we one denominator of $3$3, and the other with $6$6. We need to find a common denominator. Let's choose $6$6, as it is a common multiple of both $3$3 and $6$6.

**Do**: Change the first fraction to have a denominator of $6$6.

$\frac{2y}{3}$2y3 |
$=$= |
$\frac{2y\times2}{3\times2}$2 |

$=$= | $\frac{4y}{6}$4y6 |

So this means our expression now becomes:

$\frac{2y}{3}+\frac{5}{6}=\frac{4y}{6}+\frac{5}{6}$2`y`3+56=4`y`6+56

Success, now we can add the fractions as the denominators are common.

**Do**: Now write the numerators as a single expression above the common denominator.

$\frac{4y}{6}+\frac{5}{6}=\frac{4y+5}{6}$4`y`6+56=4`y`+56

**Reflect**: Is this simplified enough? As the terms $4y$4`y` and $5$5 are not like terms, yes, this is simplified as much as we can.

Our final example is where we have different denominators, and some simplification to perform at the final step.

$\frac{3x}{4}+\frac{3x}{2}$3`x`4+3`x`2

**Think**: Our first goal is to have common denominators. Looking at the denominators we have, $4$4 and $2$2, we can see that $4$4 is a common multiple. So use $4$4.

**Do**:

$\frac{3x}{4}+\frac{3x}{2}$3x4+3x2 |
$=$= | $\frac{3x}{4}+\frac{3x\times2}{2\times2}$3x4+3x×22×2 |

$=$= | $\frac{3x}{4}+\frac{6x}{4}$3x4+6x4 |

**Think**: Now we have a common denominator, we write the fraction as a single expression over the common denominator and then simplify where we can.

**Do**:

$\frac{3x}{4}+\frac{6x}{4}$3x4+6x4 |
$=$= | $\frac{3x+6x}{4}$3x+6x4 |

$=$= | $\frac{9x}{4}$9x4 |

**Reflect**: We collected the like terms of $3x$3`x` and $6x$6`x`. Are there any other common terms? Are they any common factors with the $9x$9`x` and $4$4? No, so this is a simplified as this answer gets.

Simplify the following:

$\frac{6x}{2}-\frac{7x}{2}$6`x`2−7`x`2

Simplify the following: $\frac{3x}{5}-\frac{x}{7}$3`x`5−`x`7

Simplify the expression $\frac{11x}{14}+\frac{7x}{21}$11`x`14+7`x`21.

It will be useful to factor to find the LCD when the denominator is more complex. Let's have a look at some examples to see how factoring the denominator can help us. In the following examples, we can use the various factoring techniques we have learned previously to determine the common denominator for each question.

Simplify the following expression, giving your answer in fully factored form:

$\frac{x}{x^2-16}-\frac{12}{x+4}$`x``x`2−16−12`x`+4

Simplify the following expression:

$\frac{x+8}{x^2+19x+88}+\frac{x-7}{\left(x+11\right)\left(x+7\right)}$`x`+8`x`2+19`x`+88+`x`−7(`x`+11)(`x`+7)

Factor the denominators and then simplify:

$\frac{4}{x^2-15x+54}-\frac{3}{x^2-36}$4`x`2−15`x`+54−3`x`2−36

Interpret expressions that represent a quantity in terms of its context.

Interpret parts of an expression, such as terms, factors, and coefficients.

Interpret complicated expressions by viewing one or more of their parts as a single entity.

Use the structure of an expression to identify ways to rewrite it. For example, to factor 3x(x − 5) + 2(x − 5), students should recognize that the "x − 5" is common to both expressions being added, so it simplifies to (3x + 2)(x − 5); or see x^4 − y^4 as (x2)^2 − (y2)^2, thus recognizing it as a difference of squares that can be factored as (x^2 − y^2)(x^2 + y^2).

Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.