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Factoring by Grouping Cont’d – Simplifying Rational Expressions

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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


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