5.1 Factoring

1
5.1 Factoring the Greatest Common Factor

• Finding the Greatest Common Factor
• Factor write each number in factored form.
• List common factors
• Choose the smallest exponents for variables and
  prime factors
• Multiply the primes and variables from step 3
• Always factor out the GCF first when factoring an
  expression

2
5.1 Factoring the Greatest Common Factor

• Example factor 5x2y 25xy2z

3
5.1 Factoring Factor By Grouping

• Factoring by grouping
• Group Terms collect the terms in 2 groups that
  have a common factor
• Factor within groups
• Factor the entire polynomial factor out a
  common binomial factor from step 2
• If necessary rearrange terms if step 3 didnt
  work, repeat steps 2 3 until you get 2 binomial
  factors

4
5.1 Factoring Factor By Grouping

• ExampleThis arrangement doesnt work.
• Rearrange and try again

5
5.2 Factoring Trinomials

• Factoring x2 bx c (no ax2 term yet)Find 2
  integers product is c and sum is b
• Both integers are positive if b and c are
  positive
• Both integers are negative if c is positive and b
  is negative
• One integer is positive and one is negative if c
  is negative

6
5.2 Factoring Trinomials

• Example
• Example

7
5.3 Factoring Trinomials Factor By Grouping

• Factoring ax2 bx c by grouping
• Multiply a times c
• Find a factorization of the number from step 1
  that also adds up to b
• Split bx into these two factors multiplied by x
• Factor by grouping (always works)

8
5.3 Factoring Trinomials Factor By Grouping

• Example
• Split up and factor by grouping

9
5.3 More on Factoring Trinomials

• Factoring ax2 bx c by using FOIL (in reverse)
• The first terms must give a product of ax2 (pick
  two)
• The last terms must have a product of c (pick
  two)
• Check to see if the sum of the outer and inner
  products equals bx
• Repeat steps 1-3 until step 3 gives a sum bx

10
5.3 More on Factoring Trinomials

• Example

11
5.3 More on Factoring Trinomials

• Box Method (not in book)

12
5.3 More on Factoring Trinomials

• Box Method keep guessing until cross-product
  terms add up to the middle value

13
5.4 Special Factoring Rules

• Difference of 2 squares
• Example
• Note the sum of 2 squares (x2 y2) cannot be
  factored.

14
5.4 Special Factoring Rules

• Perfect square trinomials
• Examples

15
5.4 Special Factoring Rules

• Difference of 2 cubes
• Example

16
5.4 Special Factoring Rules

• Sum of 2 cubes
• Example

17
5.4 Special Factoring Rules

• Summary of Factoring
• Factor out the greatest common factor
• Count the terms
• 4 terms try to factor by grouping
• 3 terms check for perfect square trinomial. If
  not a perfect square, use general factoring
  methods
• 2 terms check for difference of 2 squares,
  difference of 2 cubes, or sum of 2 cubes
• Can any factors be factored further?

18
5.5 Solving Quadratic Equations by Factoring

• Quadratic Equation
• Zero-Factor PropertyIf a and b are real numbers
  and if ab0then either a 0 or b 0

19
5.5 Solving Quadratic Equations by Factoring

• Solving a Quadratic Equation by factoring
• Write in standard form all terms on one side of
  equal sign and zero on the other
• Factor (completely)
• Set all factors equal to zero and solve the
  resulting equations
• (if time available) check your answers in the
  original equation

20
5.5 Solving Quadratic Equations by Factoring

• Example

21
5.6 Applications of Quadratic Equations

• This section covers applications in which
  quadratic formulas arise.Example Pythagorean
  theorem for right triangles (see next slide)

22
5.6 Applications of Quadratic Equations

• Pythagorean Theorem In a right triangle, with
  the hypotenuse of length c and legs of lengths a
  and b, it follows that c2 a2 b2

c
a
b
23
5.6 Applications of Quadratic Equations

• Example

x2
x
x1