University:
Glendale Community College
Course:
MAT 140 | College Mathematics
Academic year:

2019
Views:

154

Pages:

14
Author:

hugdeigury7

Section 5.5: Factoring Sums and Differences of Squares Chapter 5: Factoring Difference of Squares After any GCF is factored out, an expression is considered a difference of squares provided: • • • • Only has 2 terms Minus sign between the terms Variables have even exponents, such as 𝑥 2 , 𝑦 4 , 𝑧 6 , 𝑤 8 … Numbers are perfect squares, such as 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 … If all 4 above are true after any GCF is factored out, then we factor as follows: Step 1: Factor out GCF, if applicable Step 2: Use the template: (+)(−) Step 3: Fill in the firsts by transforming the first term as follows. • • Square root any numbers in the first term. (if applicable) Divide the exponent of the first term’s variable by 2. (if applicable) Step 4: Fill in the lasts by transforming the second term as follows • • Square root any numbers in the last term. (if applicable) Divide the exponent of the last term’s variable by 2 (if applicable). Step 5: Check Section 5.5: Factoring Sums and Differences of Squares For Example: Factor 9𝑥 2 − 16 Step 1: Factor out GCF if applicable Step 2: Use the template: (+)(−) Step 3: Fill in the firsts by transforming the first term as follows. • • Square root any numbers in the first term. Divide the exponent of the variable of the first term by 2. Step 4: Fill in the lasts by transforming the second term as follows • • Divide the exponent of the last term by 2 (if applicable). Square root any numbers in the last term. Step 5: Check Chapter 5: Factoring The problem meets all the criteria to qualify as a difference of squares. • • • • Only has 2 terms ✓ Minus sign between the terms ✓ Variables have even exponents, such as 𝑥 2 ✓ Numbers are perfect squares, such as ✓1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 … Step 1: There is no GCF, step may be skipped. Step 2: Use the template: (+)(−) Step 3: Firsts: √9 = 3 𝑥 2⁄2 = 𝑥 (3𝑥 +)(3𝑥 −) Step 4: Lasts: √16 = 4 Answer: (3𝑥 + 4)(3𝑥 − 4) Step 5: Check (3𝑥 + 4)(3𝑥 − 4) = 9𝑥 2 − 12𝑥 + 12𝑥 − 16 = 9𝑥 2 − 16 ✓ Firsts: (3𝑥)(3𝑥) = 9𝑥 2 Outers: (3𝑥)(−4) = −12𝑥 Inners: 4(3𝑥) = 12𝑥 Lasts: 4(−4) = −16 Section 5.5: Factoring Sums and Differences of Squares Chapter 5: Factoring Sums of Squares After the any GCF has factored out, an expression is considered a sum of squares provided: • • • • Only has 2 terms Plus sign between the terms Variables have even exponents, such as 𝑥 2 , 𝑦 4 , 𝑧 6 , 𝑤 8 … Numbers are perfect squares, such as 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 … If all 4 above are true, then we factor as follows: Step 1: Factor out GCF, if applicable Step 2: Stop and write your answer: Sums of Squares are prime. For Example: Factor 𝑦 4 + 25 First check to see if the problem is a sum of squares: An expression is considered a sum of squares provided: This is a sum of squares. Follow the steps: If all 4 above are true, then we factor as follows: Step 1: Factor out GCF if applicable Step 2: Stop and write your answer: Sums of Squares are prime. • • • • Only has 2 terms ✓ Plus sign between the terms✓ Variables have even exponents, 𝑦 4 ✓ Numbers are perfect squares, 25 = 52 ✓ Step 1: There is no GCF, skip this step. Step 2: Answer: Prime (sums of squares are prime) Section 5.5: Factoring Sums and Differences of Squares Factoring Chapter 5: #1- 42: Completely factor the binomials, remember to factor out the GCF first when applicable (if a problem is prime say so). 1) 𝑥 2 – 9 2) 𝑦 2 – 81 3) 𝑥 2 + 9 4) 𝑦 2 + 81 5) 𝑦 2 – 36 6) 𝑏 2 – 49 7) 𝑥 2 + 36 8) 𝑥 2 + 49 9) 25𝑎2 – 81 10) 16𝑏 2 – 49 11) 25𝑎2 + 81 12) 16𝑏 2 + 49 13) 49𝑥 2 – 36 14) 81𝑥 2 – 25 15) 49𝑥 2 + 36 16) 81𝑥 2 + 49 17) 𝑥 3 – 64𝑥 18) 𝑥 5 – 121𝑥 3 19) 𝑥 3 + 64𝑥 20) 𝑥 5 + 121𝑥 3 21) 3𝑥 2 – 27 22) 2𝑏 2 – 98 23) 3𝑥 2 + 27 24) 2𝑏 2 + 98 25) −25𝑥 2 + 9 26) −16𝑦 2 + 49 27) −16𝑥 2 + 81 28) −144𝑥 2 + 121 29) 𝑥 4 – 9 30) 𝑥 4 – 25 31) 16𝑥 4 – 25 32) 81𝑥 4 – 16 33) 98𝑥 2 – 2𝑦 4 34) 50𝑥 4 – 128𝑦 2 35) 𝑥 4 – 16 36) 𝑦 4 – 81 37) 2𝑥 4 – 512 38) 2𝑥 4 – 1250 39) 𝑦 4 – 2401 40) 𝑦 4 – 4096 41) 𝑥 4 + 4 42) 𝑦 2 + 9