Section 5.2. Factoring by Grouping

Section 5.2: Factoring by Grouping Chapter 5: Factoring Factoring by grouping • • Is a technique that is used to factor polynomials that have exactly 4 terms Is not generally used if a polynomial has more or less than 4 terms. 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 (factor) parenthesis. If this step does not produce a common (factor) parenthesis, you have made a mistake. Step 3: Factor out the GCF (this will be the answer.) Step 4: Check Section 5.2: Factoring by Grouping Factor 𝑥 3 + 𝑥 2 + 2𝑥 + 2 using 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 out the GCF (this will be the answer.) Step 4: Check Chapter 5: Factoring Step 1: Form two groups of two terms. 𝑥3 + 𝑥2 | + 2𝑥 + 2 Step 2: Factor the GCF from each pair. • • 𝑥 3 + 𝑥 2 GCF is 𝑥 2 = 𝑥 2 (𝑥 + 1) 2𝑥 + 2 GCF is 2 = 2(𝑥 + 1) =𝑥 2 (𝑥 + 1) + 2(𝑥 + 1) Step 3: The GCF is the common parenthesis of (𝑥 + 1) = (𝒙 + 𝟏)(𝒙𝟐 + 𝟐) (This is the answer.) Step 4: Check I need to confirm that my answer equals the original problem when simplified. (𝑥 + 1)(𝑥 2 + 2) Firsts: 𝑥𝑥 2 = 𝑥 3 Outers (𝑥)2 = 2𝑥 Inners 1𝑥 2 = 𝑥 2 Lasts 1(2) = 2 (𝑥 + 1)(𝑥 2 + 2) = 𝑥 3 + 2𝑥 + 𝑥 2 + 2 Now write in standard form: = 𝑥 3 + 𝑥 2 + 2𝑥 + 2 ✓ Section 5.2: Factoring by Grouping Factor x2 – 3x – 5x + 15 using 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 out the GCF (this will be the answer.) Step 4: Check Chapter 5: Factoring Step 1: Form two groups of two terms. x2 – 3x | – 5x + 15 Step 2: Factor the GCF from each pair. • • 𝑥 2 − 3𝑥 GCF is x = 𝑥(𝑥 − 3) −5𝑥 + 15 (keep the negative, and factor out the opposite of the GCF GCF is 5 Opposite of GCF is −5 = −5(𝑥 − 3) =𝑥(𝑥 − 3) − 5(𝑥 − 3) Step 3: The GCF is the common parenthesis of (𝑥 − 3) = (𝒙 − 𝟑)(𝒙 − 𝟓) (This is the answer.) Step 4: Check I will need to simplify my answer, and the original problem to check. My answer is correct if they equal each other when simplified. (𝑥 − 3)(𝑥 − 5) Firsts: 𝑥𝑥 = 𝑥 2 Outers (𝑥)(−5) = −5𝑥 Inners -3(x)= −3𝑥 Lasts (-3)(−5) = 15 (𝑥 − 3)(𝑥 − 5) = 𝑥 2 − 5𝑥 − 3𝑥 + 15 = 𝑥 2 − 8𝑥 + 15 Now rewrite given problem: x2 – 3x – 5x + 15 = 𝑥 2 − 8𝑥 + 15 ✓ My answer is correct because the original problem and the answer reduce to the same polynomial. #1 – 36: Factor by Grouping, state if a polynomial is prime 1) 𝑥 2 + 5𝑥 + 2𝑥 + 10 2) 𝑥 2 + 3𝑥 + 4𝑥 + 12 3) 𝑥 2 – 5𝑥 – 2𝑥 + 10 4) 𝑥 2 – 3𝑥 – 4𝑥 + 12 5) 𝑥 2 + 9𝑥 + 4𝑥 + 36 6) 𝑥 2 + 𝑥 + 36𝑥 + 36 7) 𝑥 2 – 9𝑥 – 4𝑥 + 36 8) 𝑥 2 – 𝑥 – 36𝑥 + 36 9) 𝑦 2 + 10𝑦 + 3𝑦 + 30 10) 𝑦 2 + 6𝑦 + 5𝑦 + 30 11) 𝑦 2 – 10𝑦 – 3𝑦 + 30 12) 𝑦 2 – 6𝑦 – 5𝑦 + 30 13) 5𝑥 2 + 5𝑥 + 6𝑥 + 6 14) 5𝑥 2 + 10𝑥 + 3𝑥 + 6 15) 5𝑥 2 – 5𝑥 – 6𝑥 + 6 16) 5𝑥 2 – 10𝑥 – 3𝑥 + 6 17) 4𝑦 2 + 3𝑦 + 4𝑦 + 3 18) 4𝑦 2 + 12𝑦 + 𝑦 + 3 19) 4𝑦 2 – 3𝑦 – 4𝑦 + 3 20) 4𝑦 2 – 12𝑦 – 𝑦 + 3 21) 2𝑧 2 – 2𝑧 + 7𝑧 – 7 22) 2𝑧 2 + 14𝑧 – 𝑧 – 7 23) 2𝑧 2 + 7𝑧 – 2𝑧 – 7 24) 2𝑧 2 – 14𝑧 + 𝑧 – 7 25) 𝑥 3 + 2𝑥 2 + 6𝑥 + 12 26) 𝑦 3 + 8𝑦 2 + 4𝑦 + 32 27) 𝑥 3 + 6𝑥 + 3𝑥 2 + 18 28) 𝑧 3 + 5𝑧 + 10𝑧 2 + 50 29) 𝑦 3 + 4𝑦 2 + 𝑦 + 4 30) 𝑥 3 + 4𝑥 2 + 𝑥 + 4 31) 𝑧 3 + 5𝑧 2 – 𝑧 – 5 32) 𝑥 3 + 9𝑥 2 – 𝑥 – 9 33) 5𝑥 3 + 4𝑥 2 + 10𝑥 + 8 34) 8𝑥 3 – 2𝑥 2 + 20𝑥 – 5 35) 7𝑡 3 + 5𝑡 2 + 21𝑡 + 15 36) 4𝑥 3 + 5𝑥 2 + 12𝑥 + 15