Section 5.7. a Review of all the Factoring Strategies – Mixed Up

Section 5.7: A Review of all the Factoring Strategies – Mixed Up Chapter 5: Factoring How to decide which technique to use when factoring. Step 1: Factor out the GCF. Step 2: Count the number of terms in the original problem. Problem has 4 terms – use factoring by grouping Steps for factoring by grouping: Step 1: Form two groups of two terms. • • The first group will consist of the first two terms. The second group will consist of the third and the fourth term. Step 2: Factor the GCF from each pair. • • This step must produce a common parenthesis. If this step does not produce a common parenthesis, you have made a mistake. Step 3: Factor by grouping (this will be the answer.) Step 4: Check Problem has 3 terms – determine whether there is or is not a number in front of the “𝑥 2 ” term and choose the appropriate technique. NO number in front of the "𝑥 2 " term – SITE factor Step 1: Factor out GCF when applicable. Step 2: Find all factors of the last number “c” with the appropriate signs from template. Step 3: Find the pair of factors that add to the middle number “b”. The polynomial is prime if there are no factors that add to the middle number “b”. Step 4: Use the numbers to write the answer. Step 5: Check There is a number in front of the x 2 term – rewrite with 4 terms and factor by grouping. Step 1: Factor out GCF when applicable. Step 2: Multiply “a” and “c” and find all the factors of this product with the appropriate signs from template. Step 3: Find the pair of factors that add to the middle number “b”. The polynomial is prime if there are no factors that add to the middle number “b”. Step 4: Rewrite the “middle term” using the numbers. Step 5: Rewrite the problem with 4 terms. Step 6: Factor by grouping to find the answer Step 7: Check Section 5.7: A Review of all the Factoring Strategies – Mixed Up Chapter 5: Factoring Problem has 2 terms – determine if the problem is one of the four special types we learned. 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 𝑥2, 𝑦4, 𝑧6, 𝑤 8 … Numbers are perfect squares, such as 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121 Sums of Squares After the any GCF has factored out, an expression is considered a sum of squares provided: • • • • Difference of Cubes After any GCF is factored out, an expression is considered a difference of cubes provided: • • • Only has 2 terms Minus sign between the terms Variables have exponents that are multiples of 3, such as 𝑥 3 , 𝑦 6 , 𝑧 9 , 𝑤 12 … Numbers are perfect cubes, such as 1, 8, 27, 64, 125, 216, 343, 512 Only has 2 terms Plus sign between the terms Variables have even exponents, such as 𝑥2, 𝑦4, 𝑧6, 𝑤 8 … Numbers are perfect squares, such as 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, Sum of Cubes After any GCF is factored out, an expression is considered a sum of cubes provided: • • • • Only has 2 terms Plus sign between the terms Variables have exponents that are multiples of 3, such as 𝑥 3 , 𝑦 6 , 𝑧 9 , 𝑤 12 … Numbers are perfect cubes, such as 1, 8, 27, 64, 125, 216, 343, 512, … Section 5.7: A Review of all the Factoring Strategies – Mixed Up Chapter 5: Factoring Use the appropriate factoring strategy once the problem type is determined. Difference of Squares 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 Sum of Squares Step 1: Problem is Prime, there is no additional work (assuming any GCF has been factored out.) Section 5.7: A Review of all the Factoring Strategies – Mixed Up Difference of Cubes Step 1: Factor out GCF, if applicable Step 2: Use the rule: 𝑎3 − 𝑏 3 = (𝑎 − 𝑏)(𝑎2 + 𝑎𝑏 + 𝑏 2 ) Step 3: Determine “a”. • • Cube root any numbers in the first term. (if applicable) Divide the exponent of the first term’s variable by 3. (if applicable) Step 4: Determine “b” • • Cube root any numbers in the first term. (if applicable) Divide the exponent of the first term’s variable by 3. (if applicable Step 5: Place the numbers in the formula and simplify to get the answer. Step 6: Check Chapter 5: Factoring Sum of Cubes Step 1: Factor out GCF, if applicable Step 2: Create template: 𝑎3 + 𝑏 3 = (𝑎 + 𝑏)(𝑎2 − 𝑎𝑏 + 𝑏 2 ) Step 3: Determine “a”. • • Cube root any numbers in the first term. (if applicable) Divide the exponent of the first term’s variable by 3. (if applicable) Step 4: Determine “b” • • Cube root any numbers in the first term. (if applicable) Divide the exponent of the first term’s variable by 3. (if applicable Step 5: Place the numbers in the formula and simplify to get the answer. Step 6: Check