Simplifying Rational Expression

1. RATIONAL
ALGEBRAIC
EXPRESSIONS

2. Rational Expression
A rational expression is one variable
in an expression that can be written
in the form
𝑷
𝑸
where P and Q are
polynomials in one variable and
𝑸 ≠ 𝟎.

3. Examples of rational expressions
𝟔𝒙
𝒙 𝟑 + 𝟖
𝟗𝒙
𝒂 + 𝟖
𝟐𝒎
𝟖 + 𝒃

4. SIMPLIFY THE GIVEN RAEXPRESSION
𝒙𝒚
𝒙
𝒚
𝟒𝒄 𝟑 𝒃
𝟐𝒄 𝟐 𝒃
𝟐𝒄

5. • Factor the numerator and denominator.
• Divide out the common factors. In this case,
the common factors divide to become 1.
• Write in simplified form.

6. If a, b and c are nonzero numbers,
then ac
bc
a
b
=

7. Example # 1.
9𝑥2
𝑦𝑧
12𝑥𝑦𝑧2
Step 1. Factor
the numerator
and
denominator.
3.3. 𝑥. 𝑥. 𝑦. 𝑧
3.2.2. 𝑥. 𝑦. 𝑧. 𝑧
Step 2. Divide
or cancel out
the common
factors.
Step 3. Write in
simplified form.
3𝑥
4𝑧
Final
answer
3.3. 𝑥. 𝑥. 𝑦. 𝑧
3.2.2. 𝑥. 𝑦. 𝑧. 𝑧

8. Example # 2.
3𝑥2
+ 9𝑥
12𝑥3
Step 1. Factor
the numerator
and
denominator.
3𝑥 (𝑥 + 3)
3.2.2. 𝑥. 𝑥. 𝑥
3𝑥 (𝑥 + 3)
3.2.2. 𝑥. 𝑥. 𝑥
Step 2. Divide
out the
common
factors.
Step 3. Write in
simplified form.
(𝑥 + 3)
4𝑥2
Final
answer
3𝑥2 - 3 . 𝑥 . 𝑥 = x
9𝑥 - 3 . 3. 𝑥 = 3

9. Example # 3.
𝑎 + 3
𝑎2 + 7𝑎 + 12
Step 1. Factor
the numerator
and
denominator.
𝑎 + 3
(𝑎 + 3)(𝑎 + 4)
Step 2. Divide
out the
common
factors.
Step 3. Write in
simplified form.
1
𝑎 + 4
Final
answer
𝑓𝑎𝑐𝑡𝑜𝑟𝑠 𝑜𝑓 3 𝑤ℎ𝑜𝑠𝑒 𝑠𝑢𝑚 𝑖𝑠 4:
3 𝑎𝑛𝑑 1
𝑎 + 3
(𝑎 + 3)(𝑎 + 4)