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Integrated math 3

Course: Integrated math 3 > Unit 13

Lesson 1: Cancelling common factors
Math
Integrated math 3
Rational functions
Cancelling common factors
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Simplifying rational expressions (advanced)

CCSS.Math:
HSA.APR.D
Google Classroom
Have you learned the basics of rational expression simplification? Great! Now gain more experience with some trickier examples.

What you should be familiar with before taking this lesson

A rational expression is a ratio of two polynomials. A rational expression is considered simplified if the numerator and denominator have no factors in common.
If this is new to you, we recommend that you check out our intro to simplifying rational expressions.

What you will learn in this lesson

In this lesson, you will practice simplifying more complicated rational expressions. Let's look at two examples, and then you can try some problems!

Example 1: Simplifying space, start fraction, 10, x, cubed, divided by, 2, x, squared, minus, 18, x, end fraction

Step 1: Factor the numerator and denominator
Here it is important to notice that while the numerator is a monomial, we can factor this as well.
start fraction, 10, x, cubed, divided by, 2, x, squared, minus, 18, x, end fraction, equals, start fraction, 2, dot, 5, dot, x, dot, x, squared, divided by, 2, dot, x, dot, left parenthesis, x, minus, 9, right parenthesis, end fraction
Step 2: List restricted values
From the factored form, we see that x, does not equal, 0 and x, does not equal, 9.
Step 3: Cancel common factors
2⋅5⋅x⋅x22⋅x⋅(x−9)=2⋅5⋅x⋅x22⋅x⋅(x−9)=5x2x−9\begin{aligned}\dfrac{ \tealD 2\cdot 5\cdot \purpleC{x}\cdot x^2}{ \tealD 2\cdot \purpleC{x}\cdot (x-9)}&=\dfrac{ \tealD{\cancel{ 2}}\cdot 5\cdot \purpleC{\cancel{x}}\cdot x^2}{ \tealD{\cancel{ 2}}\cdot \purpleC{\cancel{x}}\cdot (x-9)}\\ \\ &=\dfrac{5x^2}{x-9} \end{aligned}
Step 4: Final answer
We write the simplified form as follows:
start fraction, 5, x, squared, divided by, x, minus, 9, end fraction for x, does not equal, 0

Main takeaway

In this example, we see that sometimes we will have to factor monomials in order to simplify a rational expression.

Check your understanding

1) Simplify start fraction, 6, x, squared, divided by, 12, x, start superscript, 4, end superscript, minus, 9, x, cubed, end fraction.
Choose 1 answer:

Example 2: Simplifying space, start fraction, left parenthesis, 3, minus, x, right parenthesis, left parenthesis, x, minus, 1, right parenthesis, divided by, left parenthesis, x, minus, 3, right parenthesis, left parenthesis, x, plus, 1, right parenthesis, end fraction

Step 1: Factor the numerator and denominator
While it does not appear that there are any common factors, x, minus, 3 and 3, minus, x are related. In fact, we can factor minus, 1 out of the numerator to reveal a common factor of x, minus, 3.
=(3−x)(x−1)(x−3)(x+1)=−1(−3+x)(x−1)(x−3)(x+1)=−1(x−3)(x−1)(x−3)(x+1)Commutativity\begin{aligned} &\phantom{=}\dfrac{(3-x)(x-1)}{(x-3)(x+1)} \\\\ &=\dfrac{\goldD{-1}{(-3+x)}(x-1)}{{(x-3)}(x+1)} \\\\ &=\dfrac{{-1}{\tealD{(x-3)}}(x-1)}{{\tealD{(x-3)}}(x+1)}\quad\small{\gray{\text{Commutativity}}} \end{aligned}
Step 2: List restricted values
From the factored form, we see that x, does not equal, 3 and x, does not equal, minus, 1.
Step 3: Cancel common factors
=−1(x−3)(x−1)(x−3)(x+1)=−1(x−3)(x−1)(x−3)(x+1)=−1(x−1)x+1=1−xx+1\begin{aligned} &\phantom{=}\dfrac{{-1}{\tealD{(x-3)}}(x-1)}{{\tealD{(x-3)}}(x+1)}\\\\\\ &=\dfrac{{-1}{\tealD{\cancel{(x-3)}}}(x-1)}{{\tealD{\cancel{(x-3)}}}(x+1)} \\\\ &=\dfrac{-1(x-1)}{x+1} \\\\ &=\dfrac{1-x}{x+1} \end{aligned}
The last step of multiplying the minus, 1 into the numerator wasn't necessary, but it is common to do so.
Step 4: Final answer
We write the simplified form as follows:
start fraction, 1, minus, x, divided by, x, plus, 1, end fraction for x, does not equal, 3

Main takeaway

The factors x, minus, 3 and 3, minus, x are opposites since minus, 1, dot, left parenthesis, x, minus, 3, right parenthesis, equals, 3, minus, x.
In this example, we saw that these factors canceled, but that a factor of minus, 1 was added. In other words, the factors x, minus, 3 and 3, minus, x canceled to start text, negative, 1, end text.
In general opposite factors a, minus, b and b, minus, a will cancel to minus, 1 provided that a, does not equal, b.

Check your understanding

2) Simplify start fraction, left parenthesis, x, minus, 2, right parenthesis, left parenthesis, x, minus, 5, right parenthesis, divided by, left parenthesis, 2, minus, x, right parenthesis, left parenthesis, x, plus, 5, right parenthesis, end fraction.
Choose 1 answer:

3) Simplify start fraction, 15, minus, 10, x, divided by, 8, x, cubed, minus, 12, x, squared, end fraction.
for x, does not equal
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Let's try some more problems

4) Simplify start fraction, 3, x, divided by, 15, x, squared, minus, 6, x, end fraction.
Choose 1 answer:

5) Simplify start fraction, 3, x, cubed, minus, 15, x, squared, plus, 12, x, divided by, 3, x, minus, 3, end fraction.
for x, does not equal
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

6) Simplify start fraction, 6, x, squared, minus, 12, x, divided by, 6, x, minus, 3, x, squared, end fraction.
Choose 1 answer:

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