Section 5.1. The Greatest Common Factor

Section 5.1: The Greatest Common Factor Chapter 5: Factoring The Greatest common factor (GCF) of a polynomial is the product of • • The largest number that divides evenly into the coefficients The smallest exponent of any variable common to all terms. One method to find the Greatest Common Factor (GCF) between two integers is to list all the factors of the integer and choose the largest factor they have in common. (There are other methods that can be used to find a GCF.) The GCF of 24 and 18 equals 6 The GCF of 9 and 20 equals 1 Find the greatest common factor • • • • • 4 2 𝑥 ,𝑥 𝑥𝑦 2 , 𝑥 2 𝑦 3 𝑥4𝑦5𝑧2, 𝑥6𝑦 (𝑥 + 2)3 , (𝑥 + 2)4 (3𝑥 − 5), (3𝑥 − 5)2 Solution The product of the smallest exponent of any variable common to all terms • • • • • 𝑥 4 , 𝑥 2 GCF 𝑥 2 𝑥𝑦 2 , 𝑥 2 𝑦 3 GCF 𝑥𝑦 2 𝑥 4 𝑦 5 𝑧 2 , 𝑥 6 𝑦 𝐺𝐶𝐹 𝑥 4 𝑦 (both terms do not have a z, there is no z in the GCF (𝑥 + 2)3 , (𝑥 + 2)4 , 𝐺𝐶𝐹 (𝑥 + 2)3 (𝑡𝑟𝑒𝑎𝑡 𝑡ℎ𝑒 𝑥 + 2 𝑎𝑠 𝑎 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒. (3𝑥 − 5), (3𝑥 − 5)2 𝐺𝐶𝐹 (3𝑥 − 5) 𝑜𝑟 𝑗𝑢𝑠𝑡 3𝑥 − 5 𝑇ℎ𝑒 𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡 𝑖𝑠 1, 𝑠𝑜 𝑡ℎ𝑒 𝑝𝑎𝑟𝑒𝑛𝑡ℎ𝑒𝑠𝑖𝑠 𝑖𝑠𝑛′ 𝑡 𝑛𝑒𝑒𝑑𝑒𝑑. Section 5.1: The Greatest Common Factor Find the Greatest common factor • 10𝑥2 , 12𝑥 Chapter 5: Factoring Solution: • 10𝑥2 , 12𝑥 2 is the largest number that divides evenly into 10 and 12. The coefficient of the GCF is 2 x is the smallest exponent of the common variable x. The variable part of the GCF is x Answer: 𝐺𝐶𝐹 = 2𝑥 Find the GCF • 14𝑥𝑦 3 , 8𝑥2 𝑦, 6𝑥𝑦 Solution • 14𝑥𝑦 3 , 8𝑥2 𝑦, 6𝑥𝑦 2 is the largest number that divides evenly into 14 and 8 and 6. The coefficient of the GCF is 2 x is the smallest exponent of the common variable x. y has the smallest exponent of the common variable y The variable part of the GCF is 𝑥𝑦 Answer: 𝐺𝐶𝐹 = 2𝑥𝑦 Section 5.1: The Greatest Common Factor Find the GCF • 𝑥 3 , 12𝑥2 , 6𝑥 5 Chapter 5: Factoring Solution • 𝑥 3 , 12𝑥2 , 6𝑥 5 1 is the largest number that divides evenly into 1 and 12 and 6. The coefficient of the GCF is 1 x has the smallest exponent of the common variable x2. The variable part of the GCF is 𝑥2 Answer: 𝐺𝐶𝐹 = 𝑥2 Find the GCF • 12(𝑥 − 3)2 , 15(𝑥 − 3)2 Solution 12(𝑥 − 3)2 , 14(𝑥 − 3)2 1 is the largest number that divides evenly into 3 and 12 and 15. The coefficient of the GCF is 3 (𝑥 − 3)2 has the smallest exponent of the common (𝑥 − 3) factors The variable part of the GCF is (𝑥 − 3)2 Answer: GCF = 3(𝑥 − 3)2 Section 5.1: The Greatest Common Factor Rewrite by factoring out the GCF • 15𝑥 2 + 24𝑥 Step 1: Find the GCF Step 2: Write the GCF in front of a parenthesis and place the original problem in the parenthesis with each term divided by the GCF Step 3: Simplify (This is the answer) Chapter 5: Factoring Solution • 15𝑥 2 + 24𝑥 Step 1: Find the GCF 3 is the largest number that divides evenly into both 15 and 24. The number part of the GCF is 3. The x terms are 𝑥 𝑎𝑛𝑑 𝑥 2 the 𝑥1 term has the lowest exponent. The variable part of the GCF is x. Step 4: Check 𝐺𝐶𝐹 = 3𝑥 Step 2: Write the GCF in front of a parenthesis and place the original problem in the parenthesis with each term divided by the GCF • 15𝑥 2 = 3𝑥 ( 3𝑥 + 24𝑥 3𝑥 ) Step 3: Simplify (This is the answer) • = 3𝑥(5𝑥 + 8) Step 4: Check 3𝑥(5𝑥 + 8) = 3𝑥(5𝑥) + (3𝑥)(8) = 15𝑥 2 + 24𝑥 ✓ Section 5.1: The Greatest Common Factor Rewrite by factoring out the GCF • 27𝑥 3 𝑦 4 − 18𝑥 2 𝑦 3 + 36𝑥𝑦 2 Step 1: Find the GCF Step 2: Write the GCF in front of a parenthesis and place the original problem in the parenthesis with each term divided by the GCF Step 3: Simplify (This is the answer) Chapter 5: Factoring Solution • 27𝑥 3 𝑦 4 − 18𝑥 2 𝑦 3 + 36𝑥𝑦 2 Step 1: Find the GCF 9 is the largest number that divides evenly into both 27, 18 and 36. The number part of the GCF is 9. Each term has a x, and 𝑥1 term has the lowest exponent. Step 4: Check Each term has a y, and 𝑦 2 term has the lowest exponent. The variable part of the GCF is 𝑥𝑦 2 . GCF= 9𝑥𝑦 2 Step 2: Write the GCF in front of a parenthesis and place the original problem in the parenthesis with each term divided by the GCF • 27𝑥 3 𝑦 4 = 9𝑥𝑦 2 ( 9𝑥𝑦 2 − 18𝑥 2 𝑦 3 9𝑥𝑦 2 + 36𝑥𝑦 2 9𝑥𝑦 2 ) Step 3: Simplify (This is the answer) • = 9𝑥𝑦 2 (3𝑥 2 𝑦 2 − 2𝑥𝑦 + 4) Step 4: Check 9𝑥𝑦 2 (3𝑥 2 𝑦 2 − 2𝑥𝑦 + 4) = 9𝑥𝑦 2 (3𝑥 2 𝑦 2 ) + (9𝑥𝑦 2 )(−2𝑥𝑦) + 9𝑥𝑦 2 (4) = 27𝑥 3 𝑦 4 − 18𝑥 2 𝑦 3 + 36𝑥𝑦 2 ✓ Section 5.1: The Greatest Common Factor Rewrite by factoring out the OPPOSITE of the GCF • Chapter 5: Factoring Solution • −24𝑥 + 18 −24𝑥 + 18 Step 1: Find the GCF, and then the opposite of the GCF Step 2: Write the opposite of the GCF in front of a parenthesis and place the original problem in the parenthesis with each term divided by the opposite of the GCF Step 1: Find the GCF, and then opposite of the GCF 6 is the largest number that divides evenly into both 24 and 18. The number part of the GCF is 6. The second term does not have an x. There is no variable part of the GCF. Step 3: Simplify (This is the answer) GCF= 6 Step 4: Check Opposite of GCF −6 Step 2: Write the opposite of the GCF in front of a parenthesis and place the original problem in the parenthesis with each term divided by the opposite of the GCF • = −6 ( −24𝑥 −6 + 18 ) −6 Step 3: Simplify • • = −6(4𝑥 + (−3)) = −6(4𝑥 − 3) This is the answer Step 4: Check −6(4𝑥 − 3) = −6(4𝑥) − (−6)(3) = −24𝑥 − (−18) = −24𝑥 + 18 ✓ Section 5.1: The Greatest Common Factor Rewrite by factoring out the GCF Chapter 5: Factoring Step 1: 1 is the largest number that divides evenly into 1 and 12 and 7. • 12𝑥(𝑥 − 5) + 7(𝑥 − 5) The coefficient of the GCF is 1 Step 1: Find the GCF Only the first term has a single x. There is no 𝑥 𝑝𝑜𝑤𝑒𝑟 term in the GCF. Step 2: Write the GCF in front of a parenthesis and place the original problem in the parenthesis with each Both terms have an (𝑥 − 5) each has an exponent of 1. term divided by the GCF Step 3: Simplify (This is the answer) Step 4: Check The variable part of the GCF is (𝑥 − 5) Keep this parenthesis, it will be helpful. GCF = (𝑥 − 5) Step 2: 12(𝑥−5) (𝑥 − 5) ( (𝑥−5) + 7(𝑥−5) (𝑥−5) ) Step 3: Cancel the (𝑥 − 5) Answer: (𝑥 − 5)(12𝑥 + 7) Step 4: To check I need to rewrite the original problem and rewrite my answer to show they equal each other. Original problem rewrite: • 12𝑥(𝑥 − 5) + 7(𝑥 − 5) = 12𝑥(𝑥) + 12𝑥(−5) + 7(𝑥) + 7(−5) = 12𝑥 2 − 60𝑥 + 7𝑥 − 35 = 12𝑥 2 − 53𝑥 − 35 Answer Rewrite: (𝑥 − 5)(12𝑥 + 7) = 𝑥(12𝑥) + 𝑥(7) + (−5)(12𝑥) + (−5)(7) = 12𝑥2 + 7𝑥 − 60𝑥 − 35 = 12𝑥 2 − 53𝑥 − 35 The original problem and my answer equal when simplified, so my answer is correct. ✓ Section 5.1: The Greatest Common Factor Chapter 5: Factoring #1-16: Factor out the GCF. 1) 2𝑦 – 10 2) 6𝑥 – 12 3) 14𝑥 3 – 7𝑥 2 + 7𝑥 4) 25𝑦 3 – 20𝑦 2 + 5𝑦 5) 𝑏 5 – 3𝑏 4 + 𝑏 3 6) 𝑥 3 – 𝑥 2 + 9𝑥 7) 12𝑥 4 – 3𝑥 3 8) 3𝑦 3 – 𝑦 2 9) 12𝑥 2 𝑦 3 + 4𝑥 3 𝑦 10) 11) 16𝑎4 𝑏 2 – 18𝑎𝑏 3 12) 32𝑎2 𝑏 6 – 24𝑎𝑏 7 13) 12𝑥𝑦𝑧 3 – 14𝑥 2 𝑦 3 – 2𝑥𝑧 14) 15𝑥 3 𝑦𝑧 + 9𝑥 2 𝑦 3 𝑧 2 – 6𝑦𝑧 15) 16𝑟 4 𝑠𝑡 3 – 4𝑟 3 𝑠𝑡 2 + 12𝑟𝑠𝑡 16) 8𝑟 4 𝑠𝑡 3 + 6𝑟 2 𝑠 2 𝑡 – 10𝑟𝑠𝑡 10𝑥 3 𝑦 4 − 15𝑥 4 𝑦 2 #17-26: Factor out a (−) from each polynomial. 17) −𝑥 + 2 18) −3𝑥 + 5 19) −𝑥 – 3 20) −𝑥 – 4 21) −5𝑥 + 9 22) −7𝑤 + 16 23) −3𝑥 + 6𝑦 – 7 24) −2𝑥 – 3𝑦 + 9 25) −4𝑥 + 6𝑧 + 11𝑠 26) −13𝑦 + 4𝑧 – 11𝑟 #27-38: Factor each polynomial by factoring out the opposite of the GCF. 27) −4𝑥 3 – 12𝑥 2 28) −20𝑦 2 – 15𝑦 29) −12𝑥 3 + 4𝑥 2 – 8𝑥 30) -42𝑎5 + 24𝑎3 – 4𝑎 31) −3𝑧 4 + 6𝑧 3 32) −5𝑦 3 + 10𝑦 2 33) −14𝑎4 𝑏 2 – 6𝑎2 𝑏 34) −2𝑎2 𝑏 2 – 4𝑎𝑏 3 35) −8𝑥𝑦𝑧 5 – 4𝑥 2 𝑦 3 + 2𝑥𝑦𝑧 36) −15𝑥 2 𝑦𝑧 + 12𝑥 2 𝑦 3 𝑧 3 – 9𝑥𝑦𝑧 37) −16𝑟 4 𝑠𝑡 3 – 4𝑟 3 𝑠𝑡 2 – 12𝑟𝑠𝑡 2 38) −8𝑟 5 𝑠𝑡 2 – 6𝑟 3 𝑠 2 𝑡 – 10𝑟𝑠𝑡 Section 5.1: The Greatest Common Factor Chapter 5: Factoring #39 – 52: Factor out the GCF 39) 𝑥(𝑥 − 4) + 3(𝑥 − 4) 40) 𝑥(𝑥 – 5) + 2(𝑥 – 5) 41) 𝑥 2 (𝑦 – 2) – 3(𝑦 – 2) 42) 𝑥 2 (𝑥 – 4) – 5(𝑥 – 4) 43) 3𝑦(𝑧 + 1) – 4(𝑧 + 1) 44) 6𝑦(𝑧 + 7) – 5(𝑧 + 7) 45) 𝑥(3𝑥 – 4) – 2(3𝑥 – 4) 46) 𝑥(5𝑥 + 1) – 4(5𝑥 + 1) 47) 3𝑥(2𝑥 – 7) + 4(2𝑥 – 7) 48) 5𝑥(6𝑥 – 1) + 6(6𝑥 – 1) 49) 2𝑥 2 (3𝑥 – 5𝑦) – 5(3𝑥 – 5𝑦) 50) 7𝑥 2 (2𝑥 – 3𝑦) – 4(2𝑥 – 3𝑦) 51) 8𝑦(𝑦 – 5) – 9(𝑦 – 5) 52) 7𝑦(𝑦 + 4) – 2(𝑦 + 4)