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8-7 Factoring Special Cases Hubarth Algebra Perfect Squares Trinomials For every real number a and b: 𝑎2 + 2𝑎𝑏 + 𝑏2 = 𝑎 + 𝑏 𝑎 + 𝑏 = (𝑎 + 𝑏)2 𝑎2 − 2𝑎𝑏 + 𝑏2 = 𝑎 − 𝑏 𝑎 − 𝑏 = (𝑎 − 𝑏)2 EXAMPLES 𝑥 2 + 10𝑥 + 25 = 𝑥 + 5 𝑥 + 5 = (𝑥 + 5)2 𝑥 2 − 10𝑥 + 25 = 𝑥 − 5 𝑥 − 5 = (𝑥 − 5)2 To recognize perfect square the first and last terms must be perfect squares. The middle term will be two times the product of the first and last terms square root. Ex 1 Factoring Perfect Squares Factor 𝑥 2 + 8𝑥 + 16 𝑥 2 is a perfect square 16 is a perfect square Our factors are (x + 4)(x + 4) = (𝑥 + 4)2 the square root of 𝑥 2 is x the square root of 16 is 4 2(4)(x) = 8x our middle term Ex 2 Factoring Perfect Squares Factor 9𝑔2 − 12𝑔 + 4 9𝑔2 is perfect square 4 is a perfect square 9𝑔2 = 3𝑔 4 =2 2 3𝑔 2 = 12𝑔 Because the middle term is negative the two factors will be negative. 3𝑔 − 2 3𝑔 − 2 = (3𝑔 − 2)2 Difference of Squares For every real number a and b: 𝑎2 − 𝑏2 = (𝑎 + 𝑏)(𝑎 − 𝑏) EXAMPLES 𝑥 2 − 81 = (𝑥 − 9)(𝑥 + 9) 16𝑥 2 − 49 = (4𝑥 − 7)(4𝑥 + 7) Both terms need to be perfect squares. The factors are the square roots of each term Ex 3 Factor the Difference of Squares Factor 𝑥 2 − 16 𝑥 − 4 (𝑥 + 4) Ex 4 Factor the Difference of Squares Factor 4𝑥 2 − 121 (2𝑥 + 11)(2𝑥 − 11) Ex 5 Factoring Out a Common Factor Factor 10𝑥 2 − 40 Find the common factor of each term. 10 goes into both terms. 10(𝑥 2 − 4) Now, factor the difference of squares. 10(𝑥 + 2)(𝑥 − 2) Practice Factor each expression 1. 𝑛2 − 16𝑛 + 64 (𝑛 − 8)2 2. 𝑥 2 + 4𝑥 + 4 3. 4𝑡 2 − 36𝑡 + 81 (𝑥 + 2)2 (2𝑡 − 9)2 4. 4𝑥 2 − 25 5. 49𝑐 2 − 144 (2𝑥 + 5)(2𝑥 − 5) (7𝑐 − 12)(7𝑐 + 12) 6. 8𝑦 2 − 50 2(2𝑦 − 5)(2𝑦 + 5)