Chapter Five Chapter Five

The new currency of IraqThe new currency of Iraq

GCF-Greatest Common GCF-Greatest Common FactorFactor

• Greater doesn’t mean bigger. • The GCF is actually SMALLER

than the original• How to find the GCF – Break it

down & Find out what they have in common.

• Find the GCF of• 1) 6x3y & 8x2y2

• 2) 12a4b & 18a2b2c

FTPFTP

• Factor the Polynomial – Write the original polynomial as a product of other polynomials

• Circle of Factoring

Polynomial

2x2+10x

Factors

2x(x+5)

Factor

Multiply

Some Examples Some Examples • Factor This: 5x3 – 35x2 +10x• 5x3

• 35x2

• 10x

• Then Rewrite

• Factor: 21x2y3 -6xy5 +15x4y2

• You Got this down?

You tryYou try• 14a2 – 21a4b

• 27b2 +18b + 9

• 6x4y2 – 9x3y2 +12x2y4

Deez solutionsDeez solutions• 7a2(2-3a2b)

• 9(3b2 +2b +1)

• 3x2y2(2x2 -3x +4y2)

Factor out poly’sFactor out poly’s• Sometimes a polynomial can be

factored out. • 6x(x-3)+y2(x-3)

• Special Note: Perspectives• -(a-b) = -1(a-b) = -a + b = b – a

Factor till you can’t factor no Factor till you can’t factor no moremore

• Factor: ax + bx – ay – by

• Factor: 6x2 – 9x - 4xy + 6y

Now YouNow You• 2y(5x-2) – 3(2-5x)

• a2 – 3a +2ab – 6b

• 2mn2 – n + 8mn – 4

• 3xy – 9y – 12 + 4x

Sol’nsSol’ns• (5x-2)(2y+3)

• (a-3)(a+2b)

• (2mn-1)(n+4)

• (x-3)(3y+4)

Homework For the first sectionHomework For the first section• 1 thru 67 eoo• 4,14, 26,40, 46, 56, 68, 69a

• CHECK YOUR ANSWERS IN THE BACK!!!!!!!

Section 5.2Section 5.2• Factor a trinomial • General form: x2 +

bx +c • Find the b and c: • x2 + 8x +12• x2 - 7x + 12• x2 - 2x + -15

Anti-F.O.I.L.Anti-F.O.I.L.• Sums of b• Products of c• Watch them signs!• Make a table -- 1st column: Products (all

possible combos), 2nd Column: Sums• Ex: x^2 – 8x + 15• Ex2: x^2 – 9x – 36 • Ex3: x^2 + 6x - 27

Now you do itNow you do it• X^2 +9x + 20

• X^2 + 7x - 18

Soln’s NowSoln’s Now• X^2 +9x + 20• (x+4)(x+5)

• X^2 + 7x - 18• (x+9)(x-2)

GCF & FactoringGCF & Factoring• Steps to factoring• 1. See if a gcf can be factored out• 2. Factor the trinomial into 2

binomials

• Ex: 4y3-4y2-24y

HomeworkHomework• 1 thru 135 eoo• 24, 42, 62, 74, 90, 108, 124, 138

Section 5.3 Hungry for MoreSection 5.3 Hungry for More

• Section 5.2 – Polynomials in the form

of x2 + bx +c

• Section 5.3– Polynomials in the form

of ax2 + bx +c

• Such a small difference, but so much more work!

–

Trial and Error Trial and Error • Important things• 1. The sign of the constant in front

of the x term of the trinomial will determine the signs of the binomial factors. – If its positive then the binomial factors

will be both be positive or both negative– If its negative then the factors with have

opposite signs.

Steps to succeedSteps to succeed• 1. Make a chart

– Column 1: Factors of the coefficient of the x2 term (all possible combos)

– Column 2: Factors of the constant with no x term (no need to do all possible combos)

• 2. In the FOIL method do the O and I (outer and inner) with the factors in column one and two in an attempt to make the middle constant ( the one with the x attached to it)

Let me show youLet me show you• Factor: 2x2-7x+3• 1.

• 2. The O and I of Foil – See white board

Factors of 2 Factors of 3

2,1 -1, -3-3,-1

Why did you learn GCF?Why did you learn GCF?• Here’s why:

• Factor 6x3 +14x2-12x

Factor by groupingFactor by grouping• We are not doing

this. Factoring is complicated enough without having to learn how to do it multiple ways.

• I hope you are not too sad???

Now You do itNow You do it• 2x2 –x – 3

• -45y3 +12y2 +12y

AnswersAnswers• 2x2 –x – 3• (x+1)(2x-3)

• 15p3 +40p2 -80p• 5x(3x-4)(x+4)

HomeworkHomework• 1 thru 130 eoo

• 4, 20, 36, 46, 70, 88, 110, 130, 134, 138

ESD in Russia (Elderly Storage Device)

Section 5.4 “Special” Section 5.4 “Special” factoringfactoring

• These factors are “special” because when you see them, you can shortcut to the answer.

• (a+b)(a-b) = a2-b2

• Ex: x2-16 • Clue: no middle term…• x2-16 is the same as (x)2 – (4)2

• Then you know that it’s the sum and difference of the two terms x and 4.

• (x+4)(x-4) • Also note that if you cant take the square root of

the number (ie 10 or 13 etc.) then you cant do the problem.

Perfect square trinomialPerfect square trinomial• (a+b)2 = (a+b)(a+b) =

a2+ab+ab+b2=a2+2ab+b2

• Ex: 4x2 -20x +25 • Look! The first and last terms are the

perfect squares 2x and 5. • Just to be sure do the I and O of the FOIL

procedure. • 2x * 5 + 2x * 5 = 10x+10x =20x Great!

Now you just have to figure out where the negative sign goes.

• Final answer (2x-5)(2x-5).

Objective B Factor Objective B Factor CompletelyCompletely

• Ex: 4x2y2+12xy2+9y2

• Remember to Factor out the GCF first.

You try itYou try it12x3-75x

a2b-7a2-b+7

4x3+28x2-120x

You try itYou try it• 12x3-75x• 3x(2x+5)(2x-5)

• a2b-7a2-b+7• (b-7)(a+1)(a-1)

• 4x3+28x2-120x• 4x(x+10)(x-3)

Homework for section 5.4Homework for section 5.4• 1-126 eoo

• 12, 30, 44, 60, 64, 80, 102, 126, 132

Section 5.5 – The end of Section 5.5 – The end of factoringfactoring

Bugger Bugger Off!Off!

Solving equationsSolving equations• The highest exponent on a variable will give you

an idea of how many possible solutions there are for that variable.

• 4x3+28x2-120x = -19 3 possible values for x– Note this doesn’t mean there are always 3 answers.

Also, there could be doubles.

• To solve equations like the one above.– Simplify the side with all the variables– Make the equation equal to zero by adding or

subtracting the constant that’s lonely– Factor – Set each factor polynomial equal to zero and solve for

the variable

Solving EquationsSolving Equations• Principals

• Principle of Zeros– If the product of two things is zero then one of

them has to be zero (or equal to zero hint, hint)• Say I am told that

• (x-2)(x-3) = 0 – This means that x-2 and/or x-3 equals zero

• So set x-2 =0 and x-3 =0 and solve.

Examples a plentyExamples a plenty• x(x-3)=0

• 2x2-50=0

• (x-3)(x-10)=-10• *must be FOILed and unFOILed

Word Problems 1of 2 Word Problems 1of 2 • Ex: The sum of the squares of two

consecutive positive even integers is equal to 100. Find the two integers NOW!

• 1st integer: n 2nd integer: n+2• n2+(n+2)2 = 100

Wordy Problems 2 of 2Wordy Problems 2 of 2• In a fit of rage, Nirzwan chucks his new

SPRINT cell phone into a well because of the poor reception. Initially it has a speed of 4 ft/s and the well is a whopping 420 ft deep. How many seconds later will the phone hit the bottom and be forever lost in a sea of pollution, pesticides, and bacteria from cattle leavings? The equation d=vt+16t2 where d = distance traveled, t = seconds of travel, and v is initial velocity (speeeed).

Your turnYour turn• 2x(x+7) = 0

• 4x2-9 = 0

• (x+2)(x-7) = 52

• The sum of the squares of two consecutive positive integers is 61. Find ‘em

• At a protest against Bill O’reily and his show the O’reily Factor, one lady makes a rectangular protest sign that is 4 inches longer than twice the width. The area of that protest sign is 96 in2 assuming the manufacturer didn’t lie. Find the length and width of the protest sign.

Answers to Your TurnAnswers to Your Turn2x(x+7) = 0

0,-7

4x2-9 = 03/2,-3/2

(x+2)(x-7) = 52-6, 11

The sum of the squares of two consecutive positive integers is 61. Find ‘em

5,6

At a protest against Bill O’reilly and his show the O’reilly Factor, one lady makes a rectangular protest sign that is 4 inches longer than

twice the width. The area of that protest sign is 96 in2 assuming the manufacturer didn’t lie. Find the length and width of the protest

sign. L: 16in, W: 6in

Section 5.5 HomeworkSection 5.5 Homework• 1-60 eoo• & Problems 64, 67, 71, 75, 85(use

words!), 91. Pics are a good idea for these word problems so I hope to see some good pics when checking your homework.

• Please, no bad pics!