1. Factoring by Grouping Cont’d – Simplifying Rational Expressions

2. Factor by Grouping (Polynomials with 4+ terms)
Write polynomial in standard form (highest exponent to none)
Using the associative property rewrite the polynomial as two binomials that are added together.
3n3 – 12n2 + 2n – 8 becomes
(3n3 – 12n2) + (2n – 8)
Find the GCF of each binomial
3n2(n – 4) + 2(n – 4)
Notice the two still in ( ) are the same. Use that binomial and create a second binomial with the GCF’s
(n – 4) (3n2 + 2)
Multiply them back together to check your work.

3. Diagram of Factoring vs Distributing
(x + 7) (x + 7)
X2 + 14x + 49
Distribute (multiply)

4. Example 5: Factoring with Opposites
Factor 2x3 – 12x – 3x
2x3 – 12x – 3x
(2x3 – 12x2) + (18 – 3x)
Group terms.
2x2(x – 6) + 3(6 – x)
Factor out the GCF of each group.
2x2(x – 6) + 3(–1)(x – 6)
Write (6 – x) as –1(x – 6).
2x2(x – 6) – 3(x – 6)
Simplify. (x – 6) is a common factor.
(x – 6)(2x2 – 3)
Factor out (x – 6).

5. Example 6 Factor each polynomial. Check your answer.
15x2 – 10x3 + 8x – 12
Group terms.
(15x2 – 10x3) + (8x – 12)
Factor out the GCF of each group.
5x2(3 – 2x) + 4(2x – 3)
Write (2x – 3) as –1(3 – 2x).
5x2(3 – 2x) + 4(–1)(3 – 2x)
Simplify. (3 – 2x) is a common factor.
5x2(3 – 2x) – 4(3 – 2x)
(3 - 2x)(5x2 – 4)

6. Simplifying Rational Expressions

7. Factor all polynomials (top & bottom)
Rational Expression – a ratio (fraction) of two polynomials where the value of the variable cannot make the denominator = 0.
𝑝𝑜𝑙𝑦𝑛𝑜𝑚𝑖𝑎𝑙 𝑝𝑜𝑙𝑦𝑛𝑜𝑚𝑖𝑎𝑙
Factor all polynomials (top & bottom)
Cancel everything you can – remember anything over itself equals 1 and can be cancelled.
Don’t forget about opposites

8. Simplify Rational Expression – Example 1
What is the simplified version of: 𝑥 −1 5𝑥 −5 ?
Factor the denominator: 𝑥−1 5(𝑥−1)
Simplify (cancel or divide out the common factor of 𝑥−1.
Answer: 1 5

9. Example 2 3𝑥−6 𝑥 2 +𝑥−6 Factor everything possible: 3(𝑥−2) (𝑥+3)(𝑥−2)
3𝑥−6 𝑥 2 +𝑥−6
Factor everything possible:
3(𝑥−2) (𝑥+3)(𝑥−2)
Simplify/cancel/divide out common factors
3 𝑥+3