How To Factor By Grouping


 Sibyl Wilkinson
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1 Lecture Notes Factoring by the ACmethod page 1 Sample Problems 1. Completely factor each of the following. a) 4a 2 mn 15abm 2 6abmn + 10a 2 m 2 c) 162a + 162b 2ax 4 2bx 4 e) 3a 2 5a 2 b) a 2 x 3 b 2 x a 2 x + b 2 x 3 d) x 2 6x + 8 f) 4b 2 b 5 2. Solve the equation 8a + 2a 2 = One side of a rectangle is 4 ft shorter than three times the other side. Find the sides if the area is 84 ft 2. Practice Problems 1. Factor completely. a) 30x 15y + 6ax 3ay f) b 2 a + ab 2 1 k) 14x 12x b) xy y x + 1 g) 2m 2 18n 4 + 2m 2 p 2 18n 4 p 2 l) 5m 2 7mn + 2n 2 c) 6a 2 b 2 4a 2 bc 10a 2 c 2 h) a 2 x 2 a 2 y 2 + b 2 x 2 b 2 y 2 m) 29px 21p x 2 d) a 2 m + 2a 2 n b 2 m 2b 2 n i) 3x 2 2x 1 n) 2x 4 3y 4 + x 2 y 2 e) x 2 4y 2 + m 2 x 2 4m 2 y 2 j) 6x 2 5x Solve each of the following equations. a) 6x 4 x 3 = 2x 2 b) 5a = 26a c) 11p + 35p 2 = 6 3. One side of a rectangle is 4 in shorter than 3 times the other side. Find the sides of the rectangle if its area is 319 in 2.
2 Lecture Notes Factoring by the ACmethod page 2 Sample Problems  Answers 1. a) am (2n + 5m) (2a 3b) b) x a 2 + b 2 (x 1) (x + 1) c) x 2 (3 + x) (3 x) (a + b) d) (x 2) (x 4) e) (a 2) (3a + 1) f) (4b 5) (b + 1) 2. 7; ft and 14 ft Practice Problems  Answers 1. a) 3 (a + 5) (2x y) b) (x 1) (y 1) c) 2a 2 (b + c) (3b 5c) d) (a b) (a + b) (m + 2n) e) (x 2y) (x + 2y) m f) (b 1) (b + 1) (a + 1) g) 2 3n 2 m 3n 2 + m p a) h) (x y) (x + y) a 2 + b 2 i) (3x + 1) (x 1) j) (3x 1) (2x 1) k) 2 (2x + 1) (3x 5) l) (5m 2n) (m n) m) (5x 3p) (2x + 7p) n) (x y) (x + y) 2x 2 + 3y ; 0; in by 29 in b) 1 5 ; 5 c) 3 5 ; 2 7
3 Lecture Notes Factoring by the ACmethod page 3 Sample Problems  Solutions 1. Completely factor each of the following. a) 4a 2 mn 15abm 2 6abmn + 10a 2 m 2 Solution: 4a 2 mn 15abm 2 6abmn + 10a 2 m 2 = the GCF is am am (4an 15bm 6bn + 10am) = rearrange am 4an 6bn +10am 15bm = am (2n (2a 3b) + 5m (2a 3b)) = am (2n + 5m) (2a 3b) The nal answer is am (2n + 5m) (2a 3b). We check by multiplication. b) a 2 x 3 b 2 x a 2 x + b 2 x 3 Solution: am (2n + 5m) (2a 3b) = am (4an 6bn + 10am 15bm) = 4a 2 mn 6abmn + 10a 2 m 2 15abm 2 a 2 x 3 b 2 x a 2 x + b 2 x 3 = the GCF is x x a 2 x 2 b 2 a 2 + b 2 x 2 = rearrange x a 2 x 2 a 2 +b 2 x 2 b 2 = x a 2 x b 2 x 2 1 = x a 2 + b 2 x 2 1 We are not done yet since x 2 1 = x further factors via the di erence of squares theorem. The nal answer is x a 2 + b 2 (x + 1) (x 1). We check by multiplication. c) 162a + 162b 2ax 4 2bx 4 Solution: x a 2 + b 2 x 2 1 = x a 2 + b 2 x = x a 2 + b 2 (x + 1) (x 1) 162a + 162b 2ax 4 2bx 4 = the GCF is a + 81b ax 4 bx 4 = 2 81 (a + b) x 4 (a + b) = 2 81 x 4 (a + b) We are not done yet, since 81 x 4 = 9 2 x 2 2 further factors via the di erence of squares theorem x 4 (a + b) = x 2 2 (a + b) = x 2 9 x 2 (a + b) One factor still further factors: 9 x 2 = 3 2 x 2 = (3 + x) (3 x). Thus the nal answer is x 2 (3 + x) (3 x) (a We check by multiplication. = x 2 9 x 2 (a + b) = x x 2 (a + b) = x 2 (3 + x) (3 x) (a + b)
4 Lecture Notes Factoring by the ACmethod page 4 d) x 2 6x + 8 Solution: We will factor by grouping. First we conduct the "pq game". Step 1. Write two equations for pq and p + q. pq = 8 1st coe cient times 3rd coe cient p + q = 6 2nd coe cient Step 2. List all possible ways of writing 8 as a product of two natural numbers. There are only two ways, 1 8 and 2 4 Step 3. Assign negative signs to some numbers in the pairs listed above. Since the product pq is positive, p and q have to have the same sign. Since the sum p + q is negative, they both have to be negative. Thus we only need to consider 1 with 8 and 2 with 4. These pairs all multiply to 8; but their sum will only equal to 6 in one case. Clearly 2 with 4 work as p and q. Step 4. We use p and q we found to express the second term as the sum of two terms Step 5. Factor by grouping. We check by multiplication: Thus our result is correct. x 2 6x + 8 = x 2 2x 4x + 8 x 2 6x + 8 = x 2x 4x + 8 = x (x 2) 4 (x 2) = (x 4) (x 2) (x 2) (x 4) = x 2 4x 2x + 8 = x 2 6x + 8 e) 3a 2 5a 2 Solution: We will factor by grouping. First we conduct the "pq game". Step 1. Write two equations for pq and p + q. pq = 6 1st coe cient times 3rd coe cient p + q = 5 2nd coe cient Step 2. List all possible ways of writing 6 as a product of two natural numbers. There are only two ways, Step 3. Assign negative signs to some numbers in the pairs listed above. Since the product pq is negative, one of p and q must be negative, and the other must be positive. Since the sum p + q is negative, we will assign the negative sign to the larger number. Thus we only need to consider These pairs all multiply to 6; but their sum will only equal to 5 in one case. Clearly 1 with 6 work as p and q. Step 4. We use p and q we found to express the second term as the sum of two terms 3a 2 5a 2 = 3a 2 + a 6a 2
5 Lecture Notes Factoring by the ACmethod page 5 Step 5. Factor by grouping. We check by multiplication: 3a 2 5a 2 = 3a + a 6a 2 = a (3a + 1) 2 (3a + 1) = (a 2) (3a + 1) (a 2) (3a + 1) = 3a 2 + a 6a 2 = 3a 2 5a 2 Thus our result is correct. f) 4b 2 b 5 Solution: we will factor by grouping. First we conduct the "pq game". pq = 20 1st coe cient times 3rd coe cient p + q = 1 2nd coe cient We start by expressing 20 as a product of two numbers. the possible pairs are, 1 with 20, 2 with 10, and 4 with 5. Since the product pq is negative, one number must be positive, the other one must be positive.. Because the sum p + q is negative, the negative sign has to be in front of the larger number. We only need to consider 1 with 20, 2 with 10, and 4 with 5. Clearly 4 with 5 work as p and q. We use these these numbers to express the second term as the sum of two terms, and then factor by grouping. We check by multiplication: Thus our result is correct. 4b 2 b 5 = 4b + 4b 5b 5 = 4b (b + 1) 5 (b + 1) = (4b 5) (b + 1) (4b 5) (b + 1) = 4b 2 + 4b 5b 5 = 4b 2 b 5 2. Solve the equation 8a + 2a 2 = 42 Solution: since this equation is of a higher degree than 1, our only method is to reduce one side to zero, factor, and then apply the special zero property. 8a + 2a 2 = 42 subtract 42, rearrange 2a 2 + 8a 42 = 0 the GCF is 2 2 a 2 + 4a 21 = 0 We will factor a 2 + 4a 21 by grouping. First we conduct the "pq game". pq = 21 1st coe cient times 3rd coe cient p + q = 4 2nd coe cient We start by expressing 21 as a product of two numbers. the only possible pairs are, 1 with 21 and 3 with 7: Since the product pq is negative, one number must be positive, the other one must be positive. Because the sum p + q is positive, the negative sign has to be in front of the smaller number. We only need to consider 1 with 20, and 3 with 7. Clearly 3 with 7 work as p and q. We use these these numbers to express the second term as the sum of two terms, and then factor by grouping. 2 a 2 + 4a 21 = 0 2 a + 7a 3a 21 = 0 2 (a (a + 7) 3 (a + 7)) = 0 2 (a 3) (a + 7) = 0
6 Lecture Notes Factoring by the ACmethod page 6 Thus our equation is 2 (a 3) (a + 7) = 0 We now apply the special zero property. If this product is zero, then either 2 = 0 or a 3 = 0 or a + 7 = 0. We solve these equations for a. a 3 = 0 or a + 7 = 0 or 2 = 0 a = 3 or a = 7 or no solution here We check both solutions. If a = 3, then LHS = 8 (3) + 2 (3) 2 = = = 42 = RHS X If a = 7, then LHS = 8 ( 7) + 2 ( 7) 2 = 8 ( 7) = = 42 = RHS X Thus both solutions, 7 and 3 are correct. 3. One side of a rectangle is 4 ft shorter than three times the other side. Find the sides if the area is 84 ft 2. Solution: Let us denote the shorter side by x: Then the other side is 3x of the rectangle. x (3x 4) = 84 multiply out parentheses 3x 2 4x = 84 subtract 84 3x 2 4x 84 = 0 4: The equation expresses the area Because the equation is quadratic, we need to factor the lefthand side and then apply the zero property. We will factor by grouping. First we conduct the "pq game". The sum of p and q has to be the linear coe cient (the number in front of x, with its sign), so it is 4. The product of p and q has to be the product of the other coe cients, 3 ( 84) = 252. pq = 252 p + q = 4 Now we need to nd p and q. Because the product is negative, we re looking for a positive and a negative number. Becuse the sum is negtive, the larger number must carry the negative sign. In summary, we are looking for two numbers that multiply to 252 and di er by 4: (Then we place a negative sign in front of the larger one.) Since these numbers are relatively close to each other, they both must also be close to p 252: We enter p 252 into the calculator and get a decimal: p :874 So we start looking for factors of 252, starting at 15; and moving down. We soon nd 14 and 18. These are our values for p and q: We use these numbers to express the linear term: 4x = 14x 18x and factor by grouping. 3x 2 4x 84 = 0 3x + 14x 18x 84 = 0 x (3x + 14) 6 (3x + 14) = 0 (x 6) (3x + 14) = 0
7 Lecture Notes Factoring by the ACmethod page 7 We now apply the zero property. Either x 6 = 0 or 3x + 14 = 0. We solve both these equations for x. or x 6 = 0 x = 6 3x + 14 = 0 3x = 14 x = Since distances can not be negative, the second solution for x, is ruled out. Thus x = 6: Then the longer 3 side is 3 (6 ft) 4 ft = 14 ft; and so the rectangle s sides are 6 ft and 14 ft long. We check: 6 ft (14 ft) = 84 ft 2 and 14 ft = 3 (6 ft) 4 ft. Thus our solution is correct. For more documents like this, visit our page at and click on Lecture Notes. questions or comments to
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