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Chapter Five

Chapter Five. The new currency of Iraq. GCF-Greatest Common Factor. 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) 6x 3 y & 8x 2 y 2 2) 12a 4 b & 18a 2 b 2 c. FTP.

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Chapter Five

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  1. Chapter Five The new currency of Iraq

  2. GCF-Greatest Common Factor • 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

  3. FTP • Factor the Polynomial – Write the original polynomial as a product of other polynomials • Circle of Factoring Multiply Polynomial 2x2+10x Factors 2x(x+5) Factor

  4. Some Examples • Factor This: 5x3 – 35x2 +10x • 5x3 • 35x2 • 10x • Then Rewrite • Factor: 21x2y3 -6xy5 +15x4y2 • You Got this down?

  5. You try • 14a2 – 21a4b • 27b2 +18b + 9 • 6x4y2 – 9x3y2 +12x2y4

  6. Deez solutions • 7a2(2-3a2b) • 9(3b2 +2b +1) • 3x2y2(2x2 -3x +4y2)

  7. Factor 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

  8. Factor till you can’t factor no more • Factor: ax + bx – ay – by • Factor: 6x2 – 9x - 4xy + 6y

  9. Now You • 2y(5x-2) – 3(2-5x) • a2 – 3a +2ab – 6b • 2mn2 – n + 8mn – 4 • 3xy – 9y – 12 + 4x

  10. Sol’ns • (5x-2)(2y+3) • (a-3)(a+2b) • (2mn-1)(n+4) • (x-3)(3y+4)

  11. Homework For the first section • 1 thru 67 eoo • 4,14, 26,40, 46, 56, 68, 69a • CHECK YOUR ANSWERS IN THE BACK!!!!!!!

  12. Section 5.2 • Factor a trinomial • General form: x2 + bx +c • Find the b and c: • x2 + 8x +12 • x2 - 7x + 12 • x2 - 2x + -15

  13. 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

  14. Now you do it • X^2 +9x + 20 • X^2 + 7x - 18

  15. Soln’s Now • X^2 +9x + 20 • (x+4)(x+5) • X^2 + 7x - 18 • (x+9)(x-2)

  16. GCF & Factoring • Steps to factoring • 1. See if a gcf can be factored out • 2. Factor the trinomial into 2 binomials • Ex: 4y3-4y2-24y

  17. Homework • 1 thru 135 eoo • 24, 42, 62, 74, 90, 108, 124, 138

  18. Section 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!

  19. 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.

  20. Steps 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)

  21. Let me show you • Factor: 2x2-7x+3 • 1. • 2. The O and I of Foil – See white board

  22. Why did you learn GCF? • Here’s why: • Factor 6x3 +14x2-12x

  23. Factor 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???

  24. Now You do it • 2x2 –x – 3 • -45y3 +12y2 +12y

  25. Answers • 2x2 –x – 3 • (x+1)(2x-3) • 15p3 +40p2 -80p • 5x(3x-4)(x+4)

  26. Homework • 1 thru 130 eoo • 4, 20, 36, 46, 70, 88, 110, 130, 134, 138 ESD in Russia (Elderly Storage Device)

  27. Section 5.4 “Special” factoring • 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.

  28. Perfect 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).

  29. Objective B Factor Completely • Ex: 4x2y2+12xy2+9y2 • Remember to Factor out the GCF first.

  30. You try it 12x3-75x a2b-7a2-b+7 4x3+28x2-120x

  31. You 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)

  32. Homework for section 5.4 • 1-126 eoo • 12, 30, 44, 60, 64, 80, 102, 126, 132

  33. Section 5.5 – The end of factoring Bugger Off!

  34. Solving 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

  35. Solving 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.

  36. Examples a plenty • x(x-3)=0 • 2x2-50=0 • (x-3)(x-10)=-10 • *must be FOILed and unFOILed

  37. 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

  38. Wordy 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).

  39. Your 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.

  40. Answers to Your Turn 2x(x+7) = 0 0,-7 4x2-9 = 0 3/2,-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

  41. Section 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!

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