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RATIONAL
ALGEBRAIC
EXPRESSIONS
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.
Examples of rational expressions
𝟔𝒙
𝒙 𝟑 + 𝟖
𝟗𝒙
𝒂 + 𝟖
𝟐𝒎
𝟖 + 𝒃
SIMPLIFY THE GIVEN RAEXPRESSION
𝒙𝒚
𝒙
𝒚
𝟒𝒄 𝟑 𝒃
𝟐𝒄 𝟐 𝒃
𝟐𝒄
• Factor the numerator and denominator.
• Divide out the common factors. In this case,
the common factors divide to become 1.
• Write in simplified form.
If a, b and c are nonzero numbers,
then ac
bc
a
b
=
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
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
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)

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Rational expression

  • 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)