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Factoring Expressions/Techniques

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Published in: Mathematics
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This presentation is on one of the topic Factoring expression in algebra.

Hafiz M / Sharjah

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Qualification: M.Sc Mathematics

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  1. N/A
  2. Topic: Factoring Expressions/ Techniques Hafiz M. Arsian Motor City
  3. Outline: Some Basic Definitions How to find CCF Factoring by using CCF Factoring by using Grouping Factoring Polynomials by using Difference of Squares Factoring Trinomials by using Perfect Square Factoring Quadratic Factoring by using Cube formula Solving Quadratic by Quadratic Formula Real word problems
  4. Algebraic Expressions An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y) and operators (like add,subtract,multiply, and divide). Here are some algebraic expressions: a-b 3x x-a/b
  5. Polynomials A polynomial is an expression consisting of variables and coefficients which only employs the operations of addition, subtraction, multiplication, and non-negative integer exponents. An example of a polynomial of a single variable x is x2 — 4x + 7. An example in three variables is x3 + 2xyz2 — yz + 1.
  6. Some Basic Definitions Factors (either numbers or polynomials) When an integer is written as a product of integers, each of the integers in the product is a factor of the original number. When a polynomial is written as a product of polynomials, each of the polynomials in the product is a factor of the original polynomial.
  7. Factoring —Factoring a polynomial means expressing it as a product of other polynomials.
  8. Greatest Common Factor Greatest common factor — largest quantity that is a factor of all the integers or polynomials involved. Finding the GCF ofa List oflntegers or Terms 1) Prime factor the numbers. 2) Identify common prime factors. 3) Take the product of all common prime factors. If there are no common prime factors, GCF is 1.
  9. Greatest Common Factor Example Find the GCF of each list of numbers. 1) 12 d 8 3 8-2-2.2 so the GCF is 2 2) 7 and 20 7=1-7 20=2-2. 5 There are no common prime factors so the GCF is 1.
  10. Greatest Common Factor Example Find the CCF of each list of numbers. and x7 1) So the GCF is x x x = x3 2) 6x5 and 4x3 6x5 = 2 3 x x x 4X3 = 2 • 2 • X • X • X so the GCF is 2. x. x. x-
  11. Factoring Method #1 Factoring polynomials with a common monomial factor (using GCF). ** Always look for a GCF before using any other factoring method.
  12. Steps: OFind the greatest common factor (GCF), ODivide the polynomial by the GCE The quotient is the other factor, D Express the polynomial as the product of the quotient and the GCE
  13. Factoring Method #1 Example : 6C3d — 12C2d2 -k 3Cd Step GCF = 3cd S p 2: Divide by GCF 2c2 — 4cd+ 1
  14. Factoring Method #1 The answer should look like this: 6c3d— 12c2d2 + — 3cd(2c2 — 4cd + 1)
  15. Factoring Method #1 Factor these on your own looking for a GCF, 1 , X3 —k 3X2 — 12 X 5X2 — -k 35 16x3y4z — 8x2y2z3 + 12xy3z2
  16. Factoring Method #2 Factoring By Grouping for polynomials with 4 or more terms
  17. Example 1: tep 1: tep S oup : Factor out GCF from each group = (b -31b 3: Factor out GCF again Factoring Method #2 b 3b + 4b 12 - (D - 3b2)+ (4b -12) -3) +4)
  18. Example 2: Factoring Method #2 2х —16х —8х +64 = 2(х3 -8х2 -4х +32) = - 8х2)+ (-4х + 32))
  19. Factoring Method #2
  20. Factoring Method #2 Factor 90 + 15у2 - 18х - зху2 90 + 15 у 2 18х — 3ху2 = 3(30 + 5у2 = 3(5 6+ 6х — ху2) у2-6. -х)
  21. Factoring Method #3 actoring polynomials that are a difference of squares,
  22. Factoring Method #3 To actor, express each term as a square of a onomial then apply the rule...
  23. Factoring Method #3 x — 42 4)
  24. Factoring Method #3 Here is another example: x2 -81 =
  25. Try these: ye -121 9y2- 169x2 x2- 16
  26. Q: Solve the Word problems by using factoring. 1. The area of a square is numerically equal to twice its perimeter. Find the length of aside of the square. 2. The square of a number equals nine times that number. Find the number. 3. Suppose that four times the square of a number equals 20 times that number. What is the number?
  27. 4. The combined area of two squares is 20 square centimeters. Each side of one square is twice as long as a side of the other square. Find the lengths of the sides of each square. , The sum of the areas of two squares is 234 square inches. Each side of the larger square is five times the length of aside of the smaller square. Find the length of a side of each square.
  28. Factoring Method #4 actoring a trinomial in the form:
  29. Factoring Method #4 Factoring a perfect square trinomial in the form: a2 —2ab+ b2 2
  30. Factoring Method #4 Perfect Square Trinomials can be factored just like other trinomials (guess and check), but if you recognize the perfect squares pattern, follow the formula! a2 —2ab+b
  31. Ex: x a D es the middle t rm fit the attern, 2ab? Perfect Sq_Jares +16 2 b 2 4 = 8x
  32. Perfect Yes, the factors are (a + x 2 + 8x + 16 = (x +
  33. (2x)2 a Does the middle term fit the pattern, 2ab? Perfect Ex 4x2 -12x+9 2x,3 2 b = 12x
  34. Perfect S qUares Yes, the factors are (a - -3)2
  35. If the x2 term has no coefficient (other than Step 1: List all pairs of numbers that multiply to equal the constant, 12, Step 2: Choose the pair that adds up to the middle coefficient,
  36. Step 3: Fill those numbers 4) into the blanks in the binomials: x2 + 7x + 12 = (x + 3)( x + 4)
  37. Factoring Method #4 Factor. x2 + 2x - 24 This time, the constant is negative! —1 24 -2 , 12 Step 1: List all pairs of n bers that multiply to equal e constant, -24, (To get -24, one number must be positive and one negative,) Step 2: Which pair adds up to 2? Step 3: Write the binomial factors, , -24, x2 + 2x - 24 = (x - 4)( x + 6)
  38. Factoring Method #4 Factor. 3x2 + 14x + 8 This time, the x2 term DOES have a coefficient (other than 1)! 2,12 Step 1: Multiply 3 ' 8 = 24 (the leading coefficient & constant). Step 2: List all pairs of numbers that multiply to equal that product, 24, Step 3: Which pair adds up to 14? 24 1 24
  39. So then we can write them in the four terms, 3x2+ 12x+2x+8 3x+ 2)(x+ 4) 3x2 + 14x + 8 = (3x + 2)(x + 4)
  40. Special Cases Sum and Difference of Cubes: 3
  41. Special Cases Example : x 3 + 64 3 3 Rewrite as cubes Ap y the rule for sum of cubes: 3 ab + b2) —x ,4+42) — 4X -k 16)
  42. Ex: 8/ -125 = ((2y)3 - 53) Apply the rule for difference of cubes: = (2y - + 5 + (5)2) = (2 y — 514 y 2 -k I Oy -k 25
  43. Quadratic Formula The quadratic formula is used to solve any quadratic equation. tandard form of a quadratic equation is: The quadratic formula is: ac 2 a
  44. Real word Problem: Given the diagram below, approximate to the nearest foot how many feet of walking distance a person saves by cutting across the lawn instead of walking on the sidewalk. mm
  45. Given the diagram below, approximate to the nearest foot how many feet of walking distance a person saves by cutting across the lawn instead of walking on the sidewalk. The Pythagorean Theorem (X + 2) 2 + X2 = 202 X2 + 4X + 4 + X2 = 400 2X2 + 4X + 4 = 400 2x2 + 4x — 369 = 0 2(x2 + 2x — 198) = 0
  46. 2(Х2 + 2х — 198) = о 4 + 792 2 796 2
  47. 796 2 — 2 + 28.2 2 26.2 х 2 ft х=13.1 —2±28.2 2 -2-28.2 2 - 30.2 2 х--15.1
  48. x— 13.1 ft 28.2 ft 28-20= 8 ft
  49. Real Word Problems: Q 1: A ball is thrown straight up, from 3 m above the ground, with a velocity of 14 m/s. When does it hit the ground?
  50. Q 2: A Company is going to make frames as part of a new product they are launching. The frame will be cut out of a piece of steel, and to keep the weight down, the final area should be 28 cm2 The inside of the frame has to be 11 cm by 6 cm. What should the width x of the metal be. 6
  51. Two resistors are in parallel, like in this diagram: The total resistance has been measured at 2 Ohms, and one of the resistors is known to be 3 ohms re than the other. What are the values of the two resistors? The formula to work out total resistance "RT" is: I/RT = I/RI + 1/R2 RI
  52. Q 4: An object is thrown downward with an initial velocity of 19 feet per second. The distance, d it travels in an amount of time, t is given by the equation d=19t +15t2. How long does it take the object to fall 50 feet?
  53. Q 5: A 3 hour river cruise goes 15 km upstream and then back again. The river has a current of 2 km an hour. What is the boat's speed and how long was the upstream journey? IS km g hours 2 km/h
  54. HA Y OU (THANK YOOY Y OO e {Awesome People Set)

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