When factoring, we ALWAYS start with the (unless it s 1).

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1 Math 100 Elementary Algebra Sec 5.1: The Greatest Common Factor and Factor By Grouping (FBG) Recall: In the product XY, X and Y are factors. Defn In an expression, any factor that is common to each term is called a common factor. The largest of all common factors is called the greatest common factor (GCF). Remark: The answer upon factoring is always a. When factoring, we ALWAYS start with the (unless it s 1). Ex 1 Factor. (Check work by multiplying.) a) b) 21a 4 14a a 2 96x 2 y 2 144x 3 y + 48xy c) d) 6(3a + b) z(3a + b) 3c(bc 3a) 12(bc 3a) 6b(bc 3a) e) f) Factor out 1/3. 7c 2 (c + 4d) + c + 4d 2 3 x2 (2x 1) 4 x(2x 1) + 3(2x 1) 3 g) h) 20a 3 b 3 18a 3 b a 4 b 4 6x 3xy + 9y Ex 2 PP Find the area of the shaded region in factored form. R is the radius of the larger circle and r is the radius of the smaller circle. Ans: π(r 2 r 2 ) Page 1 of 13

2 Practice Problems Factor. 1) 6xy 15z ) 20x 2 32xy + 12x 3) 7(4x 5) b(4x 5) 4) 2x(8y + 3z) 5y(8y + 3z) 5) 4x 3 (x 1) (x 1) Use FBG method when factoring a polynomial with 4 (or more) terms. Ex 3 Factor. a) xy 4x + 3y 12 b) 10ab 3c + 5ac 6b c) x 2 2x xy + 2y d) 15a 3 25a 2 b 18ab b 3 e) ax + bx + cx + ay + by + cy Practice Problems Factor. 1) x 6y xy 2) 15x 9xb + 20w 12bw 3) x 3 5x 2 3x ) 7a + 21b + 2ab + 6b 2 5) 30a a 2 b 25ab 2 10b 3 Good Start?: (x 3 5x 2 ) (3x + 15) Sec 5.2: Factoring Trinomials of the Form x 2 + bx + c (and 5.7) We will dissect the FOIL method to factor trinomials. Consider different combinations of (x 2)(x 5). Observe numbers and signs. Factoring Trinomials of the Form x 2 + bx + c The factorization will have the form (x + m)(x + n) where mn = c and m + n = b. Ex 4 Factor. a) x x + 32 b) y 2 16y + 60 c) y y 60 Page 2 of 13

3 d) y 2 11y 60 e) 2x 2 + 2x + 12 f) 6x x + 18 How can we adjust? g) 3x 2 y 6xy + 21xz h) x 8 2x 4 15 i) x x PP PP j) x x k) If x 4 is a factor of x 2 + bx 20, what is the value of b? Facts About Signs Each represents some positive number. x 2 + x + will factor as (x + )(x + ) x 2 x + will factor as (x )(x ) x 2 + x will factor as (x + )(x ) x 2 x will factor as (x + )(x ) Last Sign of any Poly S D um ame signs ifference ifferent signs Sec 5.7: Solving Quadratic Equations by Factoring Defn A quadratic equation is an equation of the form ax 2 + bx + c = 0, where a, b, and c are real numbers and a 0. ax 2 + bx + c = 0 is the standard form of a quadratic equation. Zero Factor Property If a b = 0, then a = 0 or b = 0 (or both) Exs Solve for x. (x 3)(x + 2) = x(2x 1)(3x + 4) = 0 Steps to Solve a Quadratic Equation by Factoring Use ZFP 1) Make sure the equation is set to 0. 2) Factor, if possible, the quadratic expression. 3) Set each factor containing a variable equal to 0. 4) Solve the resulting equations to find each root. 5) Check each root. Page 3 of 13

4 Ex 5 Solve. a) x 2 6x 7 = 0 b) 2x 2 12x = 54 c) x(x + 1) = 110 2x(x 6) = 54 Ex 6 Find the area of the shaded region of the figure below in factored form. The dimensions of the smaller rectangle are x (x + 2) Practice Problems Factor. 1) x 2 y + 14xy + 48y 2) 2x 2 12x 54 Sec 5.3: Factoring Trinomials of the Form ax 2 + bx + c AND Sec 5.4: The Difference of Two Squares and Perfect Square Trinomials Ex 7 Factor using the trial-and-error method. a) 3x x + 10 b) 12x 2 + 5x 3 c) 4x 2 7x 15 What multiplies to 10 and adds to 11? Ex 8 4x 2 7x 15 = 0 Ex 9 Factor. a) 3x 2 13x 10 b) 12x 2 + 7x 12 Page 4 of 13

5 c) 36x 2 1 d) 25a 2 64b 2 e) 25x x x 2 + 0x 1 f) 25x 2 10x + 1 g) 9 100y 2 h) 36x i) If 2x 5 is a factor of 6x 2 + bx + 10, what is the value of b? Ex 10 #68 At the beginning of every football game, the referee flips a coin to see who will kick off. The equation that gives the height (in feet) of the coin tossed in the air is h = t 16t 2. a) Factor the equation. b) Use the factored form of the equation to find the height of the quarter after 0 seconds, 1 second, and 2 seconds. Factoring the Difference of Two Squares a 2 b 2 = (a + b)(a b) Note: a 2 + b 2 is. Perfect Square Trinomials a 2 + 2ab + b 2 = (a + b) 2 a 2 2ab + b 2 = (a b) 2 Page 5 of 13

6 Ex 11 Factor or solve. 12x 2 + x 6 12x 2 = x + 6 x 2 x + 5 = 0 Can this appear on exam 2? In-Class Problems/Quiz: How Am I Doing? 1) 3a 2 10a 8 2) 10x 2 + x 2 3) 3x 2 23x ) 4x 2 11x 3 5) 12x 2 24x + 9 6) 20x 2 38x ) 6x xy + 12y 2 8) 14x 3 20x 2 16x 9) 24x 2 98x 45 (Quiz EC) Ex 12 Factor. a) 36x xy + 25y 2 b) 121y 2 49 c) 50a 2 160ab + 128b 2 d) 5x x + 80 e) 2x 2 32x f) y 2 z 12yz + 36z g) x h) x 2 16 i) 2x 4 32 Page 6 of 13

7 Practice Problems Factor. 16x 2 36y 2 49x 2 28x x y 2 50x 2 25x x x 2 20x x 2 28xy + 4y 2 3x x 2 192x x 2 ± + 81y 2 Sec 5.6: Factoring: A General Review AND Sec 5.7: Solving Quadratic Equations by Factoring Refer to Factoring Polynomials Guide. Indicate number of terms for each type. Factor completely or solve for the roots of each quadratic equation. If the polynomial is not factorable, you must state that it s prime. Check answers! How? What will each answer look like? 14) 3ax + 9bx 12aw 36bw 15) x 3 + 2x 2 y 15xy 2 16) 8 + 7x x 2 Do DO Do 17) 4x 2 + 2x = 0 18) x 2 + x 42 19) 7x Do PP Do 20) x ) x 2 7x 14 22) 4x 2 4x 80 = 0 Do Do Do 23) 8x 3 22x 2 + 5x = 0 24) 100x ) 10x 2 + x 2 Do Do Do Page 7 of 13

8 26) 25x y 2 27) 64x x = 9 28) x 3 5x 2 4x + 20 Do Do 29) 2x 2 10x 14 30) 3x 2 33x ) 4x 4 11x 2 3 Do 32) 18x 2 69x ) 2x 2 + x ) 12x xy 5y 2 Do Do 35) 4x 2 13x 12 36) x 2 + 7x ) (x + 3y) 2 16 Do Do 16 (x + 3y) 2 Page 8 of 13

9 38) 8xw + 9x xy y 2 w + x 2 39) 10x 2 + 5xy 20 40) 18 2x 2 Start Do Ans: (5x + 4w)(2x + 7y 2 ) 41) 25x 2 = 36 42) (x 3) 4 + 4(x 3) 2 43) (3x 2) 3 3x + 2 Ex 44 Solve and check. a) Do b) Do (2x 3)(x 1) = 3 (x 5)(x + 4) = 2(x 5) c) Do d) Do e) PP x 2 + 5x 6 = 4 (3x 4)(5x + 1)(2x 7) = 0 (1119x 1)(777x + 19) = 0 Ans: x = 1, Page 9 of 13

10 f) g) h) x(12 x) = = (x + 3) 2 + x 2 4x x 2 9x 27 = 0 Ex 45 PP Grade the solution. 81x 2 16 = (9x + 4)(9x 4) = (3x + 2)(3x 2) Ex 46 Fill in the boxes to create a perfect square trinomial. 64x 2 ± 81y 2 Ex 47 Consider (29x + 7)(29x 14) = 0, (29x + 7)(29x 14) = 1, and (29x + 7)(29x 14) = x. Ex 48 Sec 5.8: Applications of Quadratic Equations (#6) The product of two consecutive odd integers is 1 less than 4 times their sum. Find the two integers. Define variable and set up. PP-solve. Ans: 7 and 9 OR -1 and 1 Page 10 of 13

11 Ex 49 (#12) One number is 2 more than twice another. Their product is 2 more than twice their sum. Find the numbers. Ex 50 (#14) The length of a rectangle is 3 more than twice the width. The area is 44 square inches. Find the dimensions. Pythagorean Theorem In any right triangle, if c is the length of the hypotenuse and a and b are the lengths of the two legs, then a 2 + b 2 = c 2. b c a Ex 51 (#18) The hypotenuse of a right triangle is 15 inches. One of the legs is 3 inches more than the other. Find the lengths of the two legs. Page 11 of 13

12 Ex 52 (#34) A rocket is fired vertically into the air with a speed of 240 feet per second. Its height at time t seconds is given by h(t) = 16t t. At what time(s) will the rocket be the following number of feet above the ground? a) PP 704 feet b) 896 feet c) Why do parts a and b have two answers? d) How long will the rocket be in the air? e) When the equation for part d is solved, one of the answers is t = 0. What does this represent? Ex 53 (#26) A company manufactures flash drives for home computers. It knows from experience that the number of drives it can sell each day, x, is related to the price p per drive by the equation x = p. At what price should the company sell the flash drives if it wants the daily revenue to be $1200? The equation for revenue is R = xp. Page 12 of 13

13 Revisit example 1a Multiplicity Ex 54 PP You are standing on the edge of a cliff near Acapulco, overlooking the ocean. The place where you stand is 180 meters from the ocean. You drop a pebble into the water. (Dropping the pebble implies that there is no initial velocity, so v = 0.) How many seconds will it take to hit the water? How far has the pebble dropped after 3 seconds? Use the formula S = 5t 2 + vt + h, where S = the height of the object v = the upward velocity in meters/second t = the time of flight in seconds h = the height above level ground from which the object is thrown Discriminant Problems from Factoring Assignment (Due: ) Page 13 of 13