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Math 100 Elementary Algebra Sec 5.1: The Greatest Common Factor and Factor By Grouping (FBG)

Ex 1 Factor. (Check work by multiplying.) a) b)

21𝑎4 − 14𝑎3 + 35𝑎2 96𝑥2𝑦2 − 144𝑥3𝑦 + 48𝑥𝑦 c) d)

6(3𝑎 + 𝑏) − 𝑧(3𝑎 + 𝑏) 3𝑐(𝑏𝑐 − 3𝑎) − 12(𝑏𝑐 − 3𝑎) − 6𝑏(𝑏𝑐 − 3𝑎) e) f) Factor out 1/3.

7𝑐2(𝑐 + 4𝑑) + 𝑐 + 4𝑑 2

3𝑥2(2𝑥 − 1) −

4

3𝑥(2𝑥 − 1) + 3(2𝑥 − 1)

g) h) 20𝑎3𝑏3 − 18𝑎3𝑏4 + 22𝑎4𝑏4 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(𝑹𝟐 − 𝒓𝟐)

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).

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Practice Problems Factor. 1) 6𝑥𝑦 − 15𝑧 + 21 2) 20𝑥2 − 32𝑥𝑦 + 12𝑥 3) 7(4𝑥 − 5) − 𝑏(4𝑥 − 5)

4) 2𝑥(8𝑦 + 3𝑧) − 5𝑦(8𝑦 + 3𝑧) 5) 4𝑥3(𝑥 − 1) − (𝑥 − 1) Ex 3 Factor. a) 𝑥𝑦 − 4𝑥 + 3𝑦 − 12 b) 10𝑎𝑏 − 3𝑐 + 5𝑎𝑐 − 6𝑏 c) 𝑥2 − 2𝑥 − 𝑥𝑦 + 2𝑦 d) 15𝑎3 − 25𝑎2𝑏 − 18𝑎𝑏2 + 30𝑏3 e) 𝑎𝑥 + 𝑏𝑥 + 𝑐𝑥 + 𝑎𝑦 + 𝑏𝑦 + 𝑐𝑦 Practice Problems Factor. 1) 18 + 3𝑥 − 6𝑦 − 𝑥𝑦 2) 15𝑥 − 9𝑥𝑏 + 20𝑤 − 12𝑏𝑤 3) 𝑥3 − 5𝑥2 − 3𝑥 + 15 4) 7𝑎 + 21𝑏 + 2𝑎𝑏 + 6𝑏2 5) 30𝑎3 + 12𝑎2𝑏 − 25𝑎𝑏2 − 10𝑏3 Good Start?: (𝒙𝟑 − 𝟓𝒙𝟐) − (𝟑𝒙 + 𝟏𝟓)

Sec 5.2: Factoring Trinomials of the Form 𝒙𝟐 + 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. Ex 4 Factor. a) 𝑥2 + 12𝑥 + 32 b) 𝑦2 − 16𝑦 + 60 c) 𝑦2 + 11𝑦 − 60

Factoring Trinomials of the Form 𝒙𝟐 + 𝒃𝒙 + 𝒄 The factorization will have the form (𝑥 + 𝑚)(𝑥 + 𝑛) where 𝑚𝑛 = 𝑐 and 𝑚 + 𝑛 = 𝑏.

Use FBG method when factoring a polynomial with 4 (or more) terms.

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d) 𝑦2 − 11𝑦 − 60 e) 2𝑥2 + 2𝑥 + 12 f) 6𝑥2 + 24𝑥 + 18 How can we adjust?

g) 3𝑥2𝑦 − 6𝑥𝑦 + 21𝑥𝑧 h) 𝑥8 − 2𝑥4 − 15 i) 𝑥2 −2

3𝑥 +

1

9

PP PP

j) 𝑥2 + 0.8𝑥 + 0.15 k) If 𝑥 − 4 is a factor of 𝑥2 + 𝑏𝑥 − 20, what is the value of 𝑏?

Sec 5.7: Solving Quadratic Equations by Factoring

Exs Solve for 𝑥.

(𝑥 − 3)(𝑥 + 2) = 0

1

2𝑥(2𝑥 − 1)(3𝑥 + 4) = 0

Facts About Signs

Each represents some positive number.

𝑥2 + 𝑥 + will factor as (𝑥 +)(𝑥 +) 𝑥2 − 𝑥 + will factor as (𝑥 −)(𝑥 −) 𝑥2 + 𝑥 − will factor as (𝑥 +)(𝑥 −) 𝑥2 − 𝑥 − will factor as (𝑥 +)(𝑥 −)

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.

Defn A quadratic equation is an equation of the form 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0, where 𝑎, 𝑏, and 𝑐 are real numbers and 𝑎 ≠ 0. 𝑎𝑥2 + 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)

Last Sign of any Poly um ame signs ifference ifferent signs

S

D

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Ex 5 Solve. a) 𝑥2 − 6𝑥 − 7 = 0 b) 2𝑥2 − 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) 𝑥2𝑦 + 14𝑥𝑦 + 48𝑦 2) 2𝑥2 − 12𝑥 − 54

Sec 5.3: Factoring Trinomials of the Form 𝒂𝒙𝟐 + 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) 3𝑥2 + 11𝑥 + 10 b) 12𝑥2 + 5𝑥 − 3 c) 4𝑥2 − 7𝑥 − 15 What multiplies to 10 and adds to 11?

Ex 8 ____________ 4𝑥2 − 7𝑥 − 15 = 0 Ex 9 Factor. a) 3𝑥2 − 13𝑥 − 10 b) 12𝑥2 + 7𝑥 − 12

12

10

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c) 36𝑥2 − 1 d) 25𝑎2 − 64𝑏2 e) 25𝑥2 + 10𝑥 + 1 𝟑𝟔𝒙𝟐 + 𝟎𝒙 − 𝟏

f) 25𝑥2 − 10𝑥 + 1 g) 9 − 100𝑦2 h) 36𝑥2 + 1 i) If 2𝑥 − 5 is a factor of 6𝑥2 + 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 = 6 + 29𝑡 − 16𝑡2. 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

𝑎2 − 𝑏2 = (𝑎 + 𝑏)(𝑎 − 𝑏) Note: 𝑎2 + 𝑏2 is ________.

Perfect Square Trinomials

𝑎2 + 2𝑎𝑏 + 𝑏2 = (𝑎 + 𝑏)2 𝑎2 − 2𝑎𝑏 + 𝑏2 = (𝑎 − 𝑏)2

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Ex 11 Factor or solve. 12𝑥2 + 𝑥 − 6 12𝑥2 = −𝑥 + 6 𝑥2 − 𝑥 + 5 = 0 Can this appear on exam 2?

In-Class Problems/Quiz: How Am I Doing?

1) 3𝑎2 − 10𝑎 − 8 2) 10𝑥2 + 𝑥 − 2 3) 3𝑥2 − 23𝑥 + 14 4) 4𝑥2 − 11𝑥 − 3 5) 12𝑥2 − 24𝑥 + 9 6) 20𝑥2 − 38𝑥 + 12 7) 6𝑥2 + 17𝑥𝑦 + 12𝑦2 8) 14𝑥3 − 20𝑥2 − 16𝑥 9) 24𝑥2 − 98𝑥 − 45 (Quiz EC)

Ex 12 Factor. a) 36𝑥2 + 60𝑥𝑦 + 25𝑦2 b) 121𝑦2 − 49 c) 50𝑎2 − 160𝑎𝑏 + 128𝑏2 d) 5𝑥2 + 40𝑥 + 80 e) 2𝑥2 − 32𝑥 + 110 f) 𝑦2𝑧 − 12𝑦𝑧 + 36𝑧 g) 𝑥2 + 16 h) 𝑥2 − 16 i) 2𝑥4 − 32

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Practice Problems Factor. 16𝑥2 − 36𝑦2 49𝑥2 − 28𝑥 + 4 100𝑥2 − 9 18𝑦2 − 50𝑥2 25𝑥2 + 20𝑥 + 4 25𝑥2 − 20𝑥 + 4 49𝑥2 − 28𝑥𝑦 + 4𝑦2 3𝑥2 − 75 72𝑥2 − 192𝑥 + 128 144𝑥2 ± ________ + 81𝑦2

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) 𝑥3 + 2𝑥2𝑦 − 15𝑥𝑦2 16) 8 + 7𝑥 − 𝑥2 Do DO Do 17) 4𝑥2 + 2𝑥 = 0 18) 𝑥2 + 𝑥 − 42 19) 7𝑥2 − 252 Do PP Do 20) 𝑥2 + 36 21) 𝑥2 − 7𝑥 − 14 22) 4𝑥2 − 4𝑥 − 80 = 0 Do Do Do 23) 8𝑥3 − 22𝑥2 + 5𝑥 = 0 24) 100𝑥2 + 25 25) 10𝑥2 + 𝑥 − 2 Do Do Do

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26) 25𝑥2 + 16𝑦2 27) 64𝑥2 + 48𝑥 = −9 28) 𝑥3 − 5𝑥2 − 4𝑥 + 20 Do Do

29) 2𝑥2 − 10𝑥 − 14 30) 3𝑥2 − 33𝑥 + 54 31) 4𝑥4 − 11𝑥2 − 3 Do 32) 18𝑥2 − 69𝑥 + 60 33) 2𝑥2 + 𝑥 + 6 34) 12𝑥2 + 11𝑥𝑦 − 5𝑦2 Do Do 35) 4𝑥2 − 13𝑥 − 12 36) 𝑥2 + 7𝑥 + 1 37) (𝑥 + 3𝑦)2 − 16 Do Do 𝟏𝟔 − (𝒙 + 𝟑𝒚)𝟐

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38) 8𝑥𝑤 + 9𝑥2 + 35𝑥𝑦2 + 28𝑦2𝑤 + 𝑥2 39) 10𝑥2 + 5𝑥𝑦 − 20 40) 18 − 2𝑥2 Start Do Ans: (𝟓𝒙 + 𝟒𝒘)(𝟐𝒙 + 𝟕𝒚𝟐) 41) 25𝑥2 = 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

𝑥2 + 5𝑥

6= 4 (3𝑥 − 4)(5𝑥 + 1)(2𝑥 − 7) = 0 (1119𝑥 − 1)(777𝑥 + 19) = 0

Ans: 𝒙 =𝟏

𝟏𝟏𝟏𝟗, −

𝟏𝟗

𝟕𝟕𝟕

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f) g) h) 𝑥(12 − 𝑥) = 32 152 = (𝑥 + 3)2 + 𝑥2 4𝑥3 + 12𝑥2 − 9𝑥 − 27 = 0 Ex 45 PP Grade the solution. 81𝑥2 − 16 = (9𝑥 + 4)(9𝑥 − 4) = (3𝑥 + 2)(3𝑥 − 2) Ex 46 Fill in the boxes to create a perfect square trinomial. 64𝑥2 ± 81𝑦2 Ex 47 Consider (29𝑥 + 7)(29𝑥 − 14) = 0, (29𝑥 + 7)(29𝑥 − 14) = 1, and (29𝑥 + 7)(29𝑥 − 14) = 𝑥.

Sec 5.8: Applications of Quadratic Equations Ex 48 (#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

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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. 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.

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 𝑎2 + 𝑏2 = 𝑐2.

𝑎

𝑏 𝑐

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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) = −16𝑡2 + 240𝑡. 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 = 800 − 100𝑝. 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.

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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 = −𝟓𝒕𝟐 + 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:________________)