introduction

2. introduction • Hello there ladies and gentlemen, boys and girls. Today we are going to learn the art of factoring. Not just any factoring but…. FACTORING BY GROUPING! Exciting right? Even better there are two different types of expressions that you can do this with and today we’re going to learn how. That’s right I'm talking about Polynomials and Trinomials ! Okay well lets get started !

3. Table of contents • SLIDE 1: Dummies Cover • SLIDE 2: Introduction Slide • SLIDE 3: Table of Contents • SLIDES 4-7: Insider Tips • SLIDES 8&9: Example Problems • SLIDES 10-14: Practice Problems • SLIDE 15: Glossary of Relevant Vocabulary • SLIDE 16: Dedication Slide

4. Insider tips • Before we get started here are some things that you should know and will help make things 110% easier about factoring. Especially since today we’re not doing just any old factoring, but factoring by grouping. I guess you could call these helpful hints from you to me, “tips.”

5. Tip #1 If the common factors left in the parentheses are opposites it’s OKAY! They are rendered as the same so you can just factor them out as if they we’re both identical, just choose which one you’d like to use and plug it in.

6. Tip#2 When trying to find factors that add up to “b” while trying to factor a trinomial by grouping, here’s a trick. Just make a t-chart. On one side put the factors, and on the other side their sums. This is an organized and simple way to make sure you don’t loose track or focus of that problem. So go ahead and try the chart! I promise it won’t fail you!

7. Tip#3 Math may seem scary but it’s actually quite easy. Just breathe, relax, and have fun. As long as you follow the instructions you can’t go wrong! Just take your time and work at a slow, and steady pace that’s comfortable for you.

8. Example 1 How to factor Polynomials by grouping: 1.If the polynomial has 4 terms then group the 2 pairs of terms. 2. Factor out the GCF for the first two terms. 3y is the GCF. 3. Factor out the GCF for the last two terms. 5 is the GCF. 4.Remove common factors from each group 5. Notice that the 2 quantities in the parentheses are now identical. That means we can factor out a common factor of (x-7). • Factor: 3xy-21y+5x-35 • = (3xy-21y) + (5x-35) -step 1 • = 3y(x-7) + 5(x-7) –step 2 • = (x-7)(3y+5) –steps 4&5 • Answer: (x-7)(3y+5)

9. How to factor Trinomials by grouping: • Find the product of ac. • ax ₂ + bx = c • 6x₂ + 17x + 10 • ac: 6(10)= 60 • 2. Find the factors of 60 that add up to 17. • ac +60 • 1,60 = 61 • 2,30 = 32 • 3,20 = 23 • 5,12 = 17 • 6,10 = 16 • 3. Write the 17x as the sum of 5x and 12x • 4. Group the 2 pairs of terms • 5. Remove common factors from each group. • 6. Notice that the 2 quantities in the parentheses are now identical. That means we can factor out a common factor of (6x+5) Example 2 Factor: 6x₂ + 17x + 10 = 6x₂ + 5x + 12x + 10 -step 3 = (6x₂ + 5x) + (12x + 10) –step 4 = x(6x + 5) + 2(6x+5) –step 5 = (x+2)(6x+ 5) –step 6 Answer: (x+2)(6x+5)

10. Practice problem #1 (trinomial) • Factor: 5x₂ + 11x + 2 • = 5x₂ + 1x + 10x + 2 • = (5x₂ + 1x ) + (10x+2) • = x(5x +1) + 2(5x+1) • Answer: (5x +1)(x+2)

11. Practice problem #2 (polynomial) • Factor: 6mx – 4m + 3rx – 2r • = (6mx-4m) + (3rx-2r) • = 2m(3x-2) + r(3x-2) • Answer: (3x-2)(2m+r)

12. Practice problem #3 (trinomial) • Factor: 4x₂ + 7x – 15 • = 4x₂ + 12x – 15x – 15 • = (4x₂ + 12x) + (-5x-15) • = 4x (x+3) + -5(x+3) • Answer: (x+3)(4x-5)

13. Practice problem #4 (polynomial) • Factor: 15x – 3xy + 4y – 20 • = (15x-3xy) + (4y-20) • = 3x(-5-y) + 4(y-5) • OPPOSITES! • Answer: (y-5)(3x+4)

14. Practice problem #5 (trinomial) • Factor: 12x₂ - 7x – 10 • = 12x₂ -15x + 8x – 10 • = (12x₂ -15x ) + (8x – 10) • = 3x(4x-5) + 2(4x-5) • Answer: (4x-5)(3x+2)

15. Glossary of relevant vocabulary • Factor: A circumstance, fact, or influence that contributes to a result or outcome. • Polynomial: one or more monomials added or subtracted • Trinomial: A polynomial with 3 terms • GCF (Greatest Common Factor): the largest number that evenly divides two numbers • Grouping: to compose an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. • Term: A mathematical expression which may form a separable part of an equation, a series, or another expression.

16. Dedication • I’d like to dedicate this power point presentation to my Mommy  . This is specially dedicated to her because I love her, and she’s a great mother that does what all good Mommy’s do… take care of their children.

18. Table of Contents • Dedication • Introduction • “Insider Tips” • 2 Example Problems • 5 Practice Problems

19. Dedication • I dedicate this to all the math students that are like me. Although you lose faith in yourself and say “I can’t do it,” don’t ever forget that you can do it. Even though its seems hard, with hard work and effort you can succeed.

20. Introduction • Radicals are quantities expressed as a root of another quantity. Basically, they are perfect squares, meaning that a number multiplied by itself equals a square root. • For Example: • 2x2 = 4 • 3x3 = 9 • 4x4 = 16 • 5x5 = 25 • 6x6 = 36 These are all perfect squares, their roots are whole numbers.

21. “Insider Tips” • 1. Try to memorize the first 10 perfect squares, they will help you solve the problems. • 2. Keep Calm, if you are not getting one of the problems, read over the steps. If you are still not getting it, move on to the next problem and then come back to it.

22. Practice Problem #1 The expression + written in simplest radical form is • (3) • (4) Step 1: Find the greatest multiple of each radical that is a square Step 2: Take the square root of that number and move it to the outside of the radical sign + + Step 3: Simplify the outside of the radical + + Step 4: Add the outside of the radicals

23. Practice Problem #2 The expression - written in simplest radical form is • (3) • (2) (4) Step 1: Find the greatest multiple of each radical that is a square - Step 2: Take the square root of that number and move it to the outside of the radical sign - Step 3: Subtract the outside of the radicals -

24. Problem #1 What is + expressed in simplest radical form? • (3) • (4) Step 1: Find the greatest multiple of each radical that is a square + Step 2: Take the square root of that number and move it to the outside of the radical sign + Step 3: Add the outside of the radicals +

25. Problem #2 • What is + expressed in simplest radical form? (1) (3) (2) (4) Step 1: Find the greatest multiple of each radical that is a square Step 2: Take the square root of that number and move it to the outside of the radical sign + + Step 3: Simplify the outside of the radical + Step 4: Add the outside of the radicals +

26. Problem #3 • What is - expressed in simplest radical form? (1) (3) (2) (4) Step 1: Find the greatest multiple of each radical that is a square Step 2: Take the square root of that number and move it to the outside of the radical sign - Step 3: Simplify the outside of the radical - - Step 4: Subtract the outside of the radicals -

27. Problem #4 • What is - expressed in simplest radical form? • (3) • (4) Step 1: Find the greatest multiple of each radical that is a square Step 2: Take the square root of that number and move it to the outside of the radical sign - - Step 3: Subtract the outside of the radicals -

28. Problem #5 • What is + expressed in simplest radical form? • (3) • (4) Step 1: Find the greatest multiple of each radical that is a square Step 2: Take the square root of that number and move it to the outside of the radical sign + + Step 3: Add the outside of the radicals +

30. Dedication Slide I would like to dedicate this project to my mom. She would always help me with my Math homework so without her none of this could’ve been possible.

31. In this powerpoint you will be learning how to divide polynomials using long division and dividing trinomials by monomials. We will be doing 2 examples together for each and then 5 examples on our own for each. It might seem hard at first but you’ll get the hang of it!

32. TABLE OF CONTENTS: • Relevant vocabulary with definitions • 1st Step-by-step example of dividing trinomials by binomials • 2nd Step-by-step example of dividing trinomials by monomials • 5 practice problems with solutions • 1st Step-by-step example of long division with polynomials • 2nd Step-by-step example of long division with polynomials • 5 practice problems with solutions

33. Vocabulary: • Polynomial: The sum or difference of terms which have variables raised to positiveintegerpowers and which have coefficients that may be real or complex. • Monomial: A polynomial with one term. • Trinomial: A polynomial with three terms which are not like terms. • Binomial: A polynomial with two terms which are not like terms. • Division: The action of separating something into parts, or the process of being separated. • Divisor: A number by which another number is to be divided • Dividend: The amount that you want to divide up. • Quotient: The result of dividing one number by another.

34. Ex.1 42x5+21x3-14x2 Steps 7x 1) put 7x under each binomial 42x5+21x3-14x2 2) divide each coefficient 7x 7x 7x 3) follow laws of exponents 6x4 +3x2 -2x

35. Ex.2 36n9-12n7+9n5 Steps: -3n3 1) Rewrite problem with -3n3 under each binomial 2) Divide coefficients 36n9-12n7+9n5 3)Follow laws of exponents -3n3 -3n3 -3n3 -12n6+4n4-3n2

36. Practice problems: • 8b5+6b3-20b 2) 24n7-36n2-12n9 2b 6n2 3) 56m12+32m9+16m4 4) 90a14-63a11+36a9 8m3 9a7 • 50x8y4+25x11y3+35x6y9 5x2y3

37. Answers to practice Problems: 8b5+6b3-20b 2) 24n7-36n2-12n9 3) 56m12+32m9+16m4 2b 6n2 8m3 8b5+6b3-20b24n7-36n2-12n956m12+32m9+16m4 2b 2b 2b 6n2 6n2 6n2 8m3 8m3 8m3 4b4+3b2-10 4n5-6-2n7 7m9+4m6+2m 4) 90a14-63a11+36a9 5) 50x8y4+25x11y3+35x6y9 9a7 5x2y3 90a14-63a11+36a950x8y4+25x11y3+35x6y9 9a7 9a7 9a7 5x2y3 5x2y3 5x2y3 10a7-7a4+4a2 10x6y+5x9+7x4y6

38. x -5 Ex.1 x+2 x2 -3x -10 Steps: x2 x(x+2) +x2 +2x 1)Divide the 1st term of the dividend by the1st term x -5x -10 of the divisor. 2) Multiply the quotient of X by both terms in -5(x+2) -5x-10 the divisor. -5x 3)Subtract your answer from the dividend. Make x 0 sure to change signs of 2nd terms. 4)Cancel out x2 and -x2. Add -3x and -2x. 5)Divide the 1st term of the dividend by 1st term of the divisor. 6)Multiply the quotient of -5 by both terms in the divisor. 7)Subtract answer from dividend and make sure Answer= (x - 5) you changed the signs of 2nd terms. 8) Cancel out -5x and 5x. Cancel out -10 and 10.

39. x + 4 Ex.2 x-2 x2 + 2x -8 Steps: X2 x(x-2)+x2 - 2x 1)Divide the 1st term of the dividend by the1st term X 4x -8 of the divisor. 2) Multiply the quotient of X by both terms in the 4x 4(x-2)+4x -8 divisor. x 3) Subtract your answer from the dividend. Make 0 sure to change signs of 2nd terms. 4) Cancel out x2 and -x2. Add 2x and 2x. 5) Divide the 1st term of the dividend by 1st term of the divisor. Answer= ( x + 4) 6) Multiply the quotient of 4 by both terms in the divisor. 7) Subtract answer from dividend and make sure you changed the signs of 2nd terms. 8) Cancel out 4x and -4x. Cancel out -8 and 8.

40. Practice problems: 1) x+1 x2 -9x -10 2) x+7 x2 + 9x + 14 3) x-2 x2 - 4 4) x-3 x2 -x -6 5) x-1 3x2 + 12x -15

41. Answers to Practice Problems: x - 10 x+2 x-2 • x+1 x2 - 9x - 10 2) x+7 x2+9x+14 3) x-2 x2-4 X2 x(x+1) +x2 +1x X2 x(x+7) +x2+7x X2 x(x-2) +x2-2x X -10x-10 X 2x+14 X -2x-4 -10x -10(x+1) -10x-10 2x 2(x+7) +2x+14 -2x -2(x-2) -2x -4 X 0 x 0 x 0 Answer= ( x - 10 ) Answer=( x + 2 ) Answer=( x - 2 ) x+2 3x+15 4) x-3 x2 - x -6 5) x-1 3x2 +12x-15 X2 x(x-3) +x2- 3x 3x2 3x(x-1) +3x2 -3x X 2x-6 x 15x-15 2x 2(x-3) +2x-6 15x 15(x-1) +15x-15 X 0 x 0 Answer=( x + 2 ) Answer=(3x + 15)

42. THE END!!!

43. Dividing Algebraic Fractions By Carissa Erickson

44. This book is dedicated to my mother and father.

45. Introduction When dividing fractions, you multiply the first fraction by the reciprocal of the second fraction. It’s the same thing when dividing two algebraic fractions.

46. Table of Contents • Cover Slide • Dedication Slide • Introduction Slide • Table of Contents • Insider Tips Slide • Vocabulary Slide • Example 1 • Example 2 • Practice Problems • Answer Slide

47. “Insider Tips” • Don’t factor the numbers that are not the same • Factor the numbers all the way • Get the answer into simplest form

48. Vocabulary • Polynomial- the sum or difference of monomials. • Monomial- a number, a variable, or the product of both, a polynomial with 1 term. • Binomial- a polynomial with 2 terms. • Trinomial- a polynomial with 3 terms. • Term- separated by addition or subtraction. • Like Terms- have the same variable parts.(includes exponents) • Coefficient- the number in front of the variable. If there is no number in front of the variable, then the number is one. • Constant- a number that stands alone. • Polynomial Division- The division of one polynomial by another, represented by an algebraic fraction and often solved by factoring. • Factoring- Rewriting an expression as the product of one or more factors. • Expression- Letters, numbers and symbols that are used to indicate an operation or series of operations involving real numbers.

49. Example 1 • 1. Change the division sign to multiplication and "flip” the second fraction. • 2. Divide the numerator & denominator by the common numbers. • 3. Multiply the remaining numbers. Answer:

50. Example 2 • ÷ 1.Change the division sign to multiplication and "flip” the second fraction. • × 2. Factor out the numbers. • × 3.Divide the numerators & denominators by the common numbers. • × 4. Multiply the remaining numbers. Answer: