7'1 The Greatest Common Factor and Factoring by Grouping

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7.1 The Greatest Common Factor and Factoring by
Grouping

• 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

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7.1 The Greatest Common Factor and Factoring by
Grouping

• Example factor 5x2y 25xy2z

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7.1 The Greatest Common Factor and Factoring 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

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7.1 The Greatest Common Factor and Factoring by
Grouping

• ExampleThis arrangement doesnt work.
• Rearrange and try again

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7.2 Factoring Trinomials of the Form x2 bx c

• 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

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7.2 Factoring Trinomials of the Form x2 bx c

• Example
• Example

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7.3 Factoring Trinomials of the Form ax2 bx c

• 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)

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7.3 Factoring Trinomials of the Form ax2 bx c

• Example
• Split up and factor by grouping

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7.3 Factoring Trinomials of the Form ax2 bx c

• 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

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7.3 Factoring Trinomials of the Form ax2 bx c

• Example

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7.3 Factoring Trinomials of the Form ax2 bx c

• Box Method (not in book)

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7.3 Factoring Trinomials of the Form ax2 bx c

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

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7.4 Factoring Binomials and Perfect Square
Trinomials

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

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7.4 Factoring Binomials and Perfect Square
Trinomials

• Perfect square trinomials
• Examples

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7.4 Factoring Binomials and Perfect Square
Trinomials

• Difference of 2 cubes
• Example

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7.4 Factoring Binomials and Perfect Square
Trinomials

• Sum of 2 cubes
• Example

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7.4 Factoring Binomials and Perfect Square
Trinomials

• 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?

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7.5 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

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7.5 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

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7.5 Solving Quadratic Equations by Factoring

• Example

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7.6 Applications of Quadratic Equations

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

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7.6 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
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7.6 Applications of Quadratic Equations

• Example

x2
x
x1
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9.3 Linear Inequalities in Two Variables

• A linear inequality in two variables can be
  written aswhere A, B, and C are real numbers
  andA and B are not zero

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9.3 Linear Inequalities in Two Variables

• Graphing a linear inequality
• Draw the graph of the boundary line.
• Choose a test point that is not on the line.
• If the test point satisfies the inequality, shade
  the side it is on, otherwise shade the opposite
  side.

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9.4 Systems of Linear Equations in Three Variables

• Linear system of equation in 3 variables
• Example

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9.4 Systems of Linear Equations in Three Variables

• Graphs of linear systems in 3 variables
• Single point (3 planes intersect at a point)
• Line (3 planes intersect at a line)
• No solution (all 3 equations are parallel planes)
• Plane (all 3 equations are the same plane)

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9.4 Systems of Linear Equations in Three Variables

• Solving linear systems in 3 variables
• Eliminate a variable using any 2 equations
• Eliminate the same variable using 2 other
  equations
• Eliminate a different variable from the equations
  obtained from (1) and (2)

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9.4 Systems of Linear Equations in Three Variables

• Solving linear systems in 3 variables
• Use the solution from (3) to substitute into 2 of
  the equations. Eliminate one variable to find a
  second value.
• Use the values of the 2 variables to find the
  value of the third variable.
• Check the solution in all original equations.