United States of America

Grades 9-12

We will use our understanding of rational numbers to extend to rational expressions in this lesson. We will combine our knowledge of polynomial division, the remainder theorem and factoring polynomials from lessons Â 3.04 Polynomial divisionÂ and Â 3.05 Factoring polynomialsÂ to rewrite rational expressions. Then we will rewrite rational functions in order to identify features of the graph, using the information we learned in Â 5.01 Rational functions.Â

- Rewrite the following improper fractions as mixed numbers: \dfrac{7}{3}, \dfrac{19}{8}, \dfrac{11}{9}
- How do you think simplifying improper fractions is related to simplifying an expression such as \dfrac{2\left(x+2\right)+5}{\left(x+2\right)}?

Recall that improper fractions can be written as mixed numbers. Given integers a and b, where both a and b are nonzero integers and a>b, the fraction \dfrac{a}{b} can be rewritten as q + \dfrac{r}{b}, where q represents a quotient and r represents a remainder, for instance:\dfrac{7}{3} = 2 + \dfrac{1}{3}

A **rational expression** is a quotient of two polynomials with a non-zero denominator. Examples of rational expressions include \dfrac{x + 1}{x - 1}, \dfrac{2}{3a + 2b}, and \dfrac{12y^2 + 4y - 1}{y^3 + 8y}, but also expressions like \dfrac{\sqrt{3}x - 2}{x} and \dfrac{5x}{1} = 5x.

Rational expressions can sometimes be simplified by factoring the numerator and denominator and then simplifying any **common factors** between them. Similar to improper fractions, rational expressions can be written as polynomial division:{\dfrac{p\left(x\right)}{b\left(x\right)}=q\left(x\right)+\dfrac{r\left(x\right)}{b\left(x\right)}}Where p\left(x\right) and b\left(x\right) are polynomials, q\left(x\right) is the quotient and r\left(x\right) is the remainder.

Recall from the previous lesson that rational functions will have domain and range restrictions due to asymptotes and points of discontinuity (holes). When rewriting and simplifying rational expressions, we need to keep in mind the restrictions on the variables to avoid undefined values in the original expressions.

Fully simplify each expression, so that the degrees of the polynomials in the numerator and denominator are as low as possible. Justify each step. State any restrictions on the variables.

a

\dfrac{2x^2 + 7x}{6x + 21}

Worked Solution

b

\dfrac{y^2 + 5y - 24}{y^3 - 27}

Worked Solution

Consider the expression \dfrac{3x-4}{x-2}.

a

Rewrite the rational expression in the form q\left(x\right) + \dfrac{r\left(x\right)}{b\left(x\right)} where r\left(x\right) is the remainder. State any restrictions on the variables.

Worked Solution

b

Graph the corresponding function of the expression you found in part (a).

Worked Solution

Idea summary

Rational expressions can sometimes be simplified by factoring the numerator and denominator and then simplifying any **common factors** between them.

Rational expressions can be written as polynomial division:{\dfrac{p\left(x\right)}{b\left(x\right)}=q\left(x\right)+\dfrac{r\left(x\right)}{b\left(x\right)}}Where p\left(x\right) and b\left(x\right) are polynomials, q\left(x\right) is the quotient and r\left(x\right) is the remainder.

Rewriting rational functions in different forms may help us identify key characteristics of their graphs.