FLC Ch 5 Math 100 Elementary Algebra Sec 5.1: The Greatest...
Transcript of FLC Ch 5 Math 100 Elementary Algebra Sec 5.1: The Greatest...
FLC Ch 5
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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)
21๐4 โ 14๐3 + 35๐2 96๐ฅ2๐ฆ2 โ 144๐ฅ3๐ฆ + 48๐ฅ๐ฆ c) d)
6(3๐ + ๐) โ ๐ง(3๐ + ๐) 3๐(๐๐ โ 3๐) โ 12(๐๐ โ 3๐) โ 6๐(๐๐ โ 3๐) e) f) Factor out 1/3.
7๐2(๐ + 4๐) + ๐ + 4๐ 2
3๐ฅ2(2๐ฅ โ 1) โ
4
3๐ฅ(2๐ฅ โ 1) + 3(2๐ฅ โ 1)
g) h) 20๐3๐3 โ 18๐3๐4 + 22๐4๐4 6๐ฅ โ 3๐ฅ๐ฆ + 9๐ฆ Ex 2 PP Find the area of the shaded region in factored form. ๐ is the radius of the larger circle and ๐ is the radius of the smaller circle. Ans: ๐ (๐น๐ โ ๐๐)
Recall: In the product ๐๐, ๐ and ๐ 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 itโs 1).
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Practice Problems Factor. 1) 6๐ฅ๐ฆ โ 15๐ง + 21 2) 20๐ฅ2 โ 32๐ฅ๐ฆ + 12๐ฅ 3) 7(4๐ฅ โ 5) โ ๐(4๐ฅ โ 5)
4) 2๐ฅ(8๐ฆ + 3๐ง) โ 5๐ฆ(8๐ฆ + 3๐ง) 5) 4๐ฅ3(๐ฅ โ 1) โ (๐ฅ โ 1) Ex 3 Factor. a) ๐ฅ๐ฆ โ 4๐ฅ + 3๐ฆ โ 12 b) 10๐๐ โ 3๐ + 5๐๐ โ 6๐ c) ๐ฅ2 โ 2๐ฅ โ ๐ฅ๐ฆ + 2๐ฆ d) 15๐3 โ 25๐2๐ โ 18๐๐2 + 30๐3 e) ๐๐ฅ + ๐๐ฅ + ๐๐ฅ + ๐๐ฆ + ๐๐ฆ + ๐๐ฆ Practice Problems Factor. 1) 18 + 3๐ฅ โ 6๐ฆ โ ๐ฅ๐ฆ 2) 15๐ฅ โ 9๐ฅ๐ + 20๐ค โ 12๐๐ค 3) ๐ฅ3 โ 5๐ฅ2 โ 3๐ฅ + 15 4) 7๐ + 21๐ + 2๐๐ + 6๐2 5) 30๐3 + 12๐2๐ โ 25๐๐2 โ 10๐3 Good Start?: (๐๐ โ ๐๐๐) โ (๐๐ + ๐๐)
Sec 5.2: Factoring Trinomials of the Form ๐๐ + ๐๐ + ๐ (and 5.7)
We will โdissectโ the FOIL method to factor trinomials. Consider different combinations of (๐ฅ 2)(๐ฅ 5). Observe numbers and signs. Ex 4 Factor. a) ๐ฅ2 + 12๐ฅ + 32 b) ๐ฆ2 โ 16๐ฆ + 60 c) ๐ฆ2 + 11๐ฆ โ 60
Factoring Trinomials of the Form ๐๐ + ๐๐ + ๐ The factorization will have the form (๐ฅ + ๐)(๐ฅ + ๐) where ๐๐ = ๐ and ๐ + ๐ = ๐.
Use FBG method when factoring a polynomial with 4 (or more) terms.
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d) ๐ฆ2 โ 11๐ฆ โ 60 e) 2๐ฅ2 + 2๐ฅ + 12 f) 6๐ฅ2 + 24๐ฅ + 18 How can we adjust?
g) 3๐ฅ2๐ฆ โ 6๐ฅ๐ฆ + 21๐ฅ๐ง h) ๐ฅ8 โ 2๐ฅ4 โ 15 i) ๐ฅ2 โ2
3๐ฅ +
1
9
PP PP
j) ๐ฅ2 + 0.8๐ฅ + 0.15 k) If ๐ฅ โ 4 is a factor of ๐ฅ2 + ๐๐ฅ โ 20, what is the value of ๐?
Sec 5.7: Solving Quadratic Equations by Factoring
Exs Solve for ๐ฅ.
(๐ฅ โ 3)(๐ฅ + 2) = 0
1
2๐ฅ(2๐ฅ โ 1)(3๐ฅ + 4) = 0
Facts About Signs
Each represents some positive number.
๐ฅ2 + ๐ฅ + will factor as (๐ฅ +)(๐ฅ +) ๐ฅ2 โ ๐ฅ + will factor as (๐ฅ โ)(๐ฅ โ) ๐ฅ2 + ๐ฅ โ will factor as (๐ฅ +)(๐ฅ โ) ๐ฅ2 โ ๐ฅ โ will factor as (๐ฅ +)(๐ฅ โ)
Steps to Solve a Quadratic Equation by Factoring โ 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 ๐๐ฅ2 + ๐๐ฅ + ๐ = 0, where ๐, ๐, and ๐ are real numbers and ๐ โ 0. ๐๐ฅ2 + ๐๐ฅ + ๐ = 0 is the standard form of a quadratic equation.
Zero Factor Property
If ๐ โ ๐ = 0, then ๐ = 0 or ๐ = 0 (or both)
Last Sign of any Poly um ame signs ifference ifferent signs
S
D
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Ex 5 Solve. a) ๐ฅ2 โ 6๐ฅ โ 7 = 0 b) 2๐ฅ2 โ 12๐ฅ = 54 c) ๐ฅ(๐ฅ + 1) = 110
๐๐(๐ โ ๐) = ๐๐
Ex 6 Find the area of the shaded region of the figure below in factored form. The dimensions of the smaller rectangle are ๐ฅ ร (๐ฅ + 2). Practice Problems Factor. 1) ๐ฅ2๐ฆ + 14๐ฅ๐ฆ + 48๐ฆ 2) 2๐ฅ2 โ 12๐ฅ โ 54
Sec 5.3: Factoring Trinomials of the Form ๐๐๐ + ๐๐ + ๐ AND
Sec 5.4: The Difference of Two Squares and Perfect Square Trinomials Ex 7 Factor using the trial-and-error method.
a) 3๐ฅ2 + 11๐ฅ + 10 b) 12๐ฅ2 + 5๐ฅ โ 3 c) 4๐ฅ2 โ 7๐ฅ โ 15 What multiplies to 10 and adds to 11?
Ex 8 ____________ 4๐ฅ2 โ 7๐ฅ โ 15 = 0 Ex 9 Factor. a) 3๐ฅ2 โ 13๐ฅ โ 10 b) 12๐ฅ2 + 7๐ฅ โ 12
12
10
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c) 36๐ฅ2 โ 1 d) 25๐2 โ 64๐2 e) 25๐ฅ2 + 10๐ฅ + 1 ๐๐๐๐ + ๐๐ โ ๐
f) 25๐ฅ2 โ 10๐ฅ + 1 g) 9 โ 100๐ฆ2 h) 36๐ฅ2 + 1 i) If 2๐ฅ โ 5 is a factor of 6๐ฅ2 + ๐๐ฅ + 10, what is the value of ๐? 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 โ = 6 + 29๐ก โ 16๐ก2. 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
๐2 โ ๐2 = (๐ + ๐)(๐ โ ๐) Note: ๐2 + ๐2 is ________.
Perfect Square Trinomials
๐2 + 2๐๐ + ๐2 = (๐ + ๐)2 ๐2 โ 2๐๐ + ๐2 = (๐ โ ๐)2
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Ex 11 Factor or solve. 12๐ฅ2 + ๐ฅ โ 6 12๐ฅ2 = โ๐ฅ + 6 ๐ฅ2 โ ๐ฅ + 5 = 0 Can this appear on exam 2?
In-Class Problems/Quiz: How Am I Doing?
1) 3๐2 โ 10๐ โ 8 2) 10๐ฅ2 + ๐ฅ โ 2 3) 3๐ฅ2 โ 23๐ฅ + 14 4) 4๐ฅ2 โ 11๐ฅ โ 3 5) 12๐ฅ2 โ 24๐ฅ + 9 6) 20๐ฅ2 โ 38๐ฅ + 12 7) 6๐ฅ2 + 17๐ฅ๐ฆ + 12๐ฆ2 8) 14๐ฅ3 โ 20๐ฅ2 โ 16๐ฅ 9) 24๐ฅ2 โ 98๐ฅ โ 45 (Quiz EC)
Ex 12 Factor. a) 36๐ฅ2 + 60๐ฅ๐ฆ + 25๐ฆ2 b) 121๐ฆ2 โ 49 c) 50๐2 โ 160๐๐ + 128๐2 d) 5๐ฅ2 + 40๐ฅ + 80 e) 2๐ฅ2 โ 32๐ฅ + 110 f) ๐ฆ2๐ง โ 12๐ฆ๐ง + 36๐ง g) ๐ฅ2 + 16 h) ๐ฅ2 โ 16 i) 2๐ฅ4 โ 32
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Practice Problems Factor. 16๐ฅ2 โ 36๐ฆ2 49๐ฅ2 โ 28๐ฅ + 4 100๐ฅ2 โ 9 18๐ฆ2 โ 50๐ฅ2 25๐ฅ2 + 20๐ฅ + 4 25๐ฅ2 โ 20๐ฅ + 4 49๐ฅ2 โ 28๐ฅ๐ฆ + 4๐ฆ2 3๐ฅ2 โ 75 72๐ฅ2 โ 192๐ฅ + 128 144๐ฅ2 ยฑ ________ + 81๐ฆ2
Sec 5.6: Factoring: A General Review AND Sec 5.7: Solving Quadratic Equations by Factoring Refer to โFactoring Polynomials Guideโ. 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 itโs prime. Check answers! How? What will each answer look like?
14) 3๐๐ฅ + 9๐๐ฅ โ 12๐๐ค โ 36๐๐ค 15) ๐ฅ3 + 2๐ฅ2๐ฆ โ 15๐ฅ๐ฆ2 16) 8 + 7๐ฅ โ ๐ฅ2 Do DO Do 17) 4๐ฅ2 + 2๐ฅ = 0 18) ๐ฅ2 + ๐ฅ โ 42 19) 7๐ฅ2 โ 252 Do PP Do 20) ๐ฅ2 + 36 21) ๐ฅ2 โ 7๐ฅ โ 14 22) 4๐ฅ2 โ 4๐ฅ โ 80 = 0 Do Do Do 23) 8๐ฅ3 โ 22๐ฅ2 + 5๐ฅ = 0 24) 100๐ฅ2 + 25 25) 10๐ฅ2 + ๐ฅ โ 2 Do Do Do
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26) 25๐ฅ2 + 16๐ฆ2 27) 64๐ฅ2 + 48๐ฅ = โ9 28) ๐ฅ3 โ 5๐ฅ2 โ 4๐ฅ + 20 Do Do
29) 2๐ฅ2 โ 10๐ฅ โ 14 30) 3๐ฅ2 โ 33๐ฅ + 54 31) 4๐ฅ4 โ 11๐ฅ2 โ 3 Do 32) 18๐ฅ2 โ 69๐ฅ + 60 33) 2๐ฅ2 + ๐ฅ + 6 34) 12๐ฅ2 + 11๐ฅ๐ฆ โ 5๐ฆ2 Do Do 35) 4๐ฅ2 โ 13๐ฅ โ 12 36) ๐ฅ2 + 7๐ฅ + 1 37) (๐ฅ + 3๐ฆ)2 โ 16 Do Do ๐๐ โ (๐ + ๐๐)๐
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38) 8๐ฅ๐ค + 9๐ฅ2 + 35๐ฅ๐ฆ2 + 28๐ฆ2๐ค + ๐ฅ2 39) 10๐ฅ2 + 5๐ฅ๐ฆ โ 20 40) 18 โ 2๐ฅ2 Start Do Ans: (๐๐ + ๐๐)(๐๐ + ๐๐๐) 41) 25๐ฅ2 = 36 42) (๐ฅ โ 3)4 + 4(๐ฅ โ 3)2 43) (3๐ฅ โ 2)3 โ 3๐ฅ + 2 Ex 44 Solve and check. a) Do b) Do
(2๐ฅ โ 3)(๐ฅ โ 1) = 3 (๐ฅ โ 5)(๐ฅ + 4) = 2(๐ฅ โ 5) c) Do d) Do e) PP
๐ฅ2 + 5๐ฅ
6= 4 (3๐ฅ โ 4)(5๐ฅ + 1)(2๐ฅ โ 7) = 0 (1119๐ฅ โ 1)(777๐ฅ + 19) = 0
Ans: ๐ =๐
๐๐๐๐, โ
๐๐
๐๐๐
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f) g) h) ๐ฅ(12 โ ๐ฅ) = 32 152 = (๐ฅ + 3)2 + ๐ฅ2 4๐ฅ3 + 12๐ฅ2 โ 9๐ฅ โ 27 = 0 Ex 45 PP Grade the solution. 81๐ฅ2 โ 16 = (9๐ฅ + 4)(9๐ฅ โ 4) = (3๐ฅ + 2)(3๐ฅ โ 2) Ex 46 Fill in the boxes to create a perfect square trinomial. 64๐ฅ2 ยฑ 81๐ฆ2 Ex 47 Consider (29๐ฅ + 7)(29๐ฅ โ 14) = 0, (29๐ฅ + 7)(29๐ฅ โ 14) = 1, and (29๐ฅ + 7)(29๐ฅ โ 14) = ๐ฅ.
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 ๐ is the length of the hypotenuse and ๐ and ๐ are the lengths of the two legs, then ๐2 + ๐2 = ๐2.
๐
๐ ๐
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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 ๐ก seconds is given by โ(๐ก) = โ16๐ก2 + 240๐ก. 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 ๐ก = 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, ๐ฅ, is related to the price ๐ per drive by the equation ๐ฅ = 800 โ 100๐. At what price should the company sell the flash drives if it wants the daily revenue to be $1200? The equation for revenue is ๐ = ๐ฅ๐.
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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 ๐ฃ = 0.) How many seconds will it take to hit the water? How
far has the pebble dropped after 3 seconds? Use the formula ๐บ = โ๐๐๐ + ๐๐ + ๐, where ๐ = the height of the object ๐ฃ = the upward velocity in meters/second ๐ก = the time of flight in seconds โ = the height above level ground from which the object is thrown
Discriminant โProblems from Factoring Assignment (Due:________________)