1. MAT 105 SPRING 2009 Factoring and Algebraic Fractions
Chapter 6
Factoring and Algebraic Fractions

2. Section 6.2 Factoring: Common Factors and Difference of Squares
MAT 105 SPRING 2009
Section 6.2 Factoring: Common Factors and Difference of Squares

3. Factoring is the reverse of multiplying.
MAT 105 SPRING 2009
Factoring is the reverse of multiplying.
A polynomial or a factor is called _________________ if it contains no factors other than 1 or -1.

4. THE FIRST STEP: Factoring Out the Greatest Common Monomial Factor
MAT 105 SPRING 2009
THE FIRST STEP: Factoring Out the Greatest Common Monomial Factor

5. Solving Formulas Involving Factoring
MAT 105 SPRING 2009
Solving Formulas Involving Factoring

6. Solving Formulas Involving Factoring
MAT 105 SPRING 2009
Solving Formulas Involving Factoring

7. Factoring the Difference of Perfect Squares
MAT 105 SPRING 2009
Factoring the Difference of Perfect Squares
Recall:
Difference of Squares:

8. Factoring the Difference of Perfect Squares
MAT 105 SPRING 2009
Factoring the Difference of Perfect Squares

9. Factor Completely: HINT: Always check for a GCF first!!
MAT 105 SPRING 2009
Factor Completely:
HINT: Always check for a GCF first!!

10. MAT 105 SPRING 2009
Factoring by Grouping (Consider grouping method if polynomial has 4 terms)
1) Always start by checking for a GCF of all 4 terms. After you factor out the GCF or if the polynomial does not have a GCF other than 1, check if the remaining 4-term polynomial can be factored by grouping.
2) Determine if you can pair up the terms in such a way that each pair has its own common factor.
3) If so, factor out the common factor from each pair.
4) If the resulting terms have a common binomial factor, factor it out.

11. MAT 105 SPRING 2009
Factor Completely

12. MAT 105 SPRING 2009
Factor Completely

13. Section 6.3 Factoring Trinomials
MAT 105 SPRING 2009
Section 6.3 Factoring Trinomials

14. Factoring Trinomials in the Form
MAT 105 SPRING 2009
Factoring Trinomials in the Form
Recall: F
O + I L
To factor a trinomial is to reverse the multiplication process (UnFOIL)

15. Before you attempt to Un-FOIL
MAT 105 SPRING 2009
Before you attempt to Un-FOIL
1) Always factor out the GCF first, if possible.
2) Write terms in descending order.
Now we begin
3) Set up the binomial factors like this: (x )(x )
4) List the factor pairs of the LAST term
*If the LAST term is POSITIVE, then the signs must be the same (both + or both -)
*If the LAST term is NEGATIVE, then the signs must be different (one + and one -).
5) Find the pair whose sum is equal to the MIDDLE term
6) Check by multiplying the binomials (FOIL)

16. MAT 105 SPRING 2009
Factor Completely

17. MAT 105 SPRING 2009
Factor Completely

18. Factoring Trinomials in the Form
MAT 105 SPRING 2009
Factoring Trinomials in the Form
The Trial & Check Method:
Before you attempt to Un-FOIL
1) Always factor out the GCF first, if possible.
2) Write terms in descending order.
Now we begin
3) Set up the binomial factors like this: ( x )( x )
4) List the factor pairs of the FIRST term
5) List the factor pairs of the LAST term
6) Sub in possible factor pairs and ‘try’ them by multiplying the binomials (FOIL) until you find the winning combination; that is when O+I =MIDDLE term.

19. MAT 105 SPRING 2009
Factor completely

20. MAT 105 SPRING 2009
Factor completely

21. MAT 105 SPRING 2009
Factor completely

22. MAT 105 SPRING 2009
Factor completely
A tricky one!

23. Section 6.4 The Sum and Difference of Cubes
MAT 105 SPRING 2009
Section The Sum and Difference of Cubes

24. The Sum and Difference of Cubes Learn these formulas!!
MAT 105 SPRING 2009
The Sum and Difference of Cubes Learn these formulas!!

25. MAT 105 SPRING 2009
Factor Completely

26. MAT 105 SPRING 2009
Factor Completely

27. A General Strategy for Factoring Polynomials
MAT 105 SPRING 2009
A General Strategy for Factoring Polynomials
Before you begin to factor, make sure the terms are written in descending
order of the exponents on one of the variables. Rearrange the terms, if necessary.
Factor out all common factors (GCF). If your leading term is negative, factor out -1.
If an expression has two terms, check for the following types of polynomials:
a) The difference of two squares: x2 - y2 = (x + y)(x - y)
b) The sum of two cubes: x3 + y3 = (x + y)(x2 - xy + y2)
c) The difference of two cubes: x3 - y3 = (x - y)(x2 + xy + y2)
If an expression has three terms, attempt to factor it as a trinomial.
If an expression has four or more terms, try factoring by grouping.
Continue factoring until each individual factor is prime. You may need to use a factoring technique more than once.
Check the results by multiplying the factors back out.

28. Section 6.5 Equivalent Fractions
MAT 105 SPRING 2009
Section Equivalent Fractions

29. MAT 105 SPRING 2009
Equivalent Fractions
The value of a fraction is unchanged if BOTH numerator and denominator are multiplied or divided by the same non-zero number.
Equivalent fractions
Equivalent fractions

30. An algebraic fraction is a ratio of two polynomials.
MAT 105 SPRING 2009
An algebraic fraction is a ratio of two polynomials.
Some examples of algebraic fractions are:
Algebraic fractions are also called rational expressions.

31. Simplifying Algebraic Fractions
MAT 105 SPRING 2009
Simplifying Algebraic Fractions
A fraction is in its simplest form if the numerator and denominator have no common factors other than 1 or -1.
(We say that the numerator and denominator are relatively prime.)
We use terms like “reduce”, “simplify”, or “put into lowest terms”.
Two simple steps for simplifying algebraic fractions:
Factor the numerator and the denominator.
2. Divide out (cancel) the common FACTORS of the numerator and the denominator.

32. Cancel only common factors.
MAT 105 SPRING 2009
WARNING:
Cancel only common factors.
Do NOT cancel terms.
Example: NEVER EVER NEVER do this!!!!!!!

33. Simplify the rational expression
MAT 105 SPRING 2009
Simplify the rational expression
Here is the plan:
Factor the numerator and the denominator.
Divide out any common factors.
Simplest form.
Notice in this example, , because the value of the denominator would be 0. ,

34. A Special Case The numerator and denominator are OPPOSITES.
MAT 105 SPRING 2009
A Special Case
The numerator and denominator are OPPOSITES.

35. MAT 105 SPRING 2009
Examples
Simplify each fraction.

36. MAT 105 SPRING 2009
Example
Simplify each fraction.

37. MAT 105 SPRING 2009
Example
Simplify each fraction.

38. Section 6.6 Multiplication and Division of Algebraic Fractions
MAT 105 SPRING 2009
Section Multiplication and Division of Algebraic Fractions

39. Multiplying Fractions
MAT 105 SPRING 2009
Multiplying Fractions
Numerical Fractions:
To multiply algebraic fractions:
Completely factor the numerator and denominator of each fraction.
Divide out common factors. (CANCEL)
Multiply the numerators and denominators of the reduced fractions:

40. Simplify the given expressions involving multiplication.
MAT 105 SPRING 2009
Simplify the given expressions involving multiplication.

41. Simplify the given expressions involving multiplication.
MAT 105 SPRING 2009
Simplify the given expressions involving multiplication.

42. Dividing Fractions To divide algebraic fractions:
MAT 105 SPRING 2009
Dividing Fractions
To divide algebraic fractions:
Invert the second fraction and multiply. (Multiply by the reciprocal of the divisor.)
Completely factor the numerator and denominator of each fraction.
Divide out common factors. (CANCEL)
Multiply the numerators and denominators of the reduced fractions:

43. Simplify the given expressions involving division.
MAT 105 SPRING 2009
Simplify the given expressions involving division.

44. Simplify the given expression involving division.
MAT 105 SPRING 2009
Simplify the given expression involving division.

45. Section 6.7 Addition and Subtraction of Algebraic Fractions
MAT 105 SPRING 2009
Section Addition and Subtraction of Algebraic Fractions

46. MAT 105 SPRING 2009
To add or subtract like fractions (fractions with the same denominator), we add/subtract the numerators and keep the denominators the same.
Example:

47. MAT 105 SPRING 2009
If the fractions do NOT have a common denominator, we will first write equivalent fractions using the Least Common Denominator.
Example:

48. Perform the operations and simplify.
MAT 105 SPRING 2009
Perform the operations and simplify.

49. Perform the operations and simplify.
MAT 105 SPRING 2009
Perform the operations and simplify.

50. Perform the operations and simplify.
MAT 105 SPRING 2009
Perform the operations and simplify.

51. Perform the operations and simplify.
MAT 105 SPRING 2009
Perform the operations and simplify.

52. Complex Fractions A fraction contained within a fraction. Example
MAT 105 SPRING 2009
Complex Fractions
A fraction contained within a fraction.
Example
To simplify a complex fraction:
Write the numerator as a single fraction.
Write the denominator as a single fraction.
Multiply the numerator by the reciprocal of the denominator.
Simplify.

53. Simplify the complex fraction.
MAT 105 SPRING 2009
Simplify the complex fraction.

54. Simplify the complex fraction.
MAT 105 SPRING 2009
Simplify the complex fraction.

55. Section 6.8 Equations Involving Algebraic Fractions
MAT 105 SPRING 2009
Section Equations Involving Algebraic Fractions

56. To solve an equation involving fractions:
MAT 105 SPRING 2009
To solve an equation involving fractions:
Multiply each term of the equation (BOTH SIDES) by the LCD to rid the equation of fractions.
Example

57. We will use the same strategy for algebraic fractions.
MAT 105 SPRING 2009
We will use the same strategy for algebraic fractions.
Factor each denominator to determine the LCD
Multiply each term of the equations by the LCD to eliminate the fractions.
Remove grouping symbols by distributing (watch out for negative signs)
Combine like terms on each side of the equation.
Solve for the variable.
Check solutions in the ORIGINAL EQUATION. (Check for extraneous solutions*.)
*If an apparent solution causes a denominator of the original equation to equal zero, we reject that answer.

58. Solve the equation and check the results.
MAT 105 SPRING 2009
Solve the equation and check the results.

59. Solve the equation and check the results.
MAT 105 SPRING 2009
Solve the equation and check the results.

60. Solve the equation and check the results.
MAT 105 SPRING 2009
Solve the equation and check the results.