Algebra 2 Rational Expressions Lesson (Free PDF)

Algebra 2 math lesson: Rational Expressions – Maths PDF

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This rational expressions lesson for Algebra 2 covers rational expressions (Common Core A-APR.D.7, A-SSE.A.2), with a free PDF to print.

A rational expression is a quotient of polynomials. To simplify, factor the numerator and denominator and cancel common factors, and state the excluded values that make the denominator zero. Multiply by multiplying across, divide by multiplying by the reciprocal, and add or subtract over a common denominator.

Algebra 2 · Common Core State Standards: A-APR.D.7, A-SSE.A.2

Key facts in this rational expressions lesson

Key facts

Rational expression = polynomial ÷ polynomial.
Factor first, then cancel common factors only.
Excluded values make the original denominator 0.
Divide by multiplying by the reciprocal.
Add or subtract over the least common denominator.

What is a rational expression?

Rational expression

A rational expression is a quotient of two polynomials, \(\frac{p(x)}{q(x)}\) with \(q(x) \ne 0\). Values of \(x\) that make the denominator zero are excluded values.

Rational expressions behave like fractions. To simplify, factor the numerator and the denominator and cancel common factors. For example, \(\frac{x^{2} – 9}{x^{2} + x – 6} = \frac{(x – 3)(x + 3)}{(x + 3)(x – 2)} = \frac{x – 3}{x – 2}\), with \(x \ne -3\) and \(x \ne 2\).

Work through rational expressions in four steps.

The excluded values come from the original denominator, even after canceling. The simplified graph keeps a hole at each canceled factor.

y = (x² – 4)/(x – 2) is the line y = x + 2 with a hole at x = 2.

How do you multiply and divide rational expressions?

Multiply numerators and denominators, factor first, and cancel. To divide, multiply by the reciprocal of the second expression. Example: \(\frac{x^{2} – 1}{x + 3} \cdot \frac{x^{2} + 3x}{x + 1} = \frac{(x – 1)(x + 1)}{x + 3} \cdot \frac{x(x + 3)}{x + 1} = x(x – 1)\).

How do you add and subtract rational expressions?

Write each expression over a common denominator, then combine the numerators. Find the least common denominator by factoring each denominator.

Factor each denominator.
Build the LCD from every factor, using the highest power.
Rewrite each fraction over the LCD.
Add or subtract the numerators and simplify.

Example: \(\frac{3}{x + 2} + \frac{5}{x – 1} = \frac{3(x – 1) + 5(x + 2)}{(x + 2)(x – 1)} = \frac{8x + 7}{(x + 2)(x – 1)}\).

Aisha paints 1/6 of the fence per hour and Diego 1/4 per hour: together 5/12.

Rational expressions model rates. If Aisha paints a fence in 6 hours and Diego paints it in 4 hours, their combined rate is \(\frac{1}{6} + \frac{1}{4} = \frac{5}{12}\) of the fence per hour, so together they need \(\frac{12}{5} = 2.4\) hours. Recognizing structure also helps: \(x^{4} – 16 = (x^{2})^{2} – 4^{2} = (x^{2} – 4)(x^{2} + 4)\).

Excluded values come from the original denominator.

Rational expressions lesson: questions and answers

How do you simplify a rational expression?

Factor the numerator and the denominator completely, then cancel common factors. State the excluded values from the original denominator.

What are excluded values?

Excluded values are the inputs that make a denominator zero. They stay excluded even if the factor cancels.

How do you add rational expressions with different denominators?

Factor the denominators, find the least common denominator, rewrite each expression over it, and add the numerators.

How do you divide rational expressions?

Multiply by the reciprocal of the divisor, then factor and cancel. For example, a/b divided by c/d equals a/b times d/c.

Can you cancel terms that are added?

No. You can cancel only common factors. In (x + 3)/3, you cannot cancel the 3s because 3 is a term in the numerator, not a factor.

How to use this rational expressions lesson

Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.

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