Section 5.4 Factoring

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Section 5.4 Factoring

• FACTORING
• Greatest Common Factor,
• Factor by Grouping,
• Factoring Trinomials,
• Difference of Squares,
• Perfect Square Trinomial,
• Sum Difference of Cubes

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Factoringdefine factored form

• Factor means to write a quantity as a
  multiplication problem
• a product of the factors.
• Factored forms of 18 are

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Factoring The Greatest Common Factor

• To find the greatest common factor of a list of
  numbers
• Write each number in prime factored form
• Choose the least amount of each prime that occurs
  in each number
• Multiply them together
• Find the GCF of 24 36

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Factoring The Greatest Common Factor

• To find the greatest common factor of a list of
  variable terms
• Choose the variables common to each term.
• Choose the smallest exponent of each common
  variable.
• Multiply the variables.
• Find the GCF of

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Factoring The Greatest Common Factor

• To factor out the greatest common factor of a
  polynomial
• Choose the greatest common factor for the
  coefficients.
• Choose the greatest common factor for the
  variable parts.
• Multiply the factors.

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Factoring The Greatest Common Factor

• Factor
• each of the
• following
• by factoring
• out the
• greatest
• common
• factor

5x 5 4ab 10a2 8p4q3 6p3q2 2y 4y2
16y3 3x(y 2) -1(y 2)
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Factoring The Greatest Common Factor

• The answers are

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Factoring by Grouping

• Often used when factoring four terms.
• Organize the terms in two groups of two terms.
• Factor out the greatest common factor from each
  group of two terms.
• Factor out the common binomial factor from the
  two groups.
• Rearranging the terms may be necessary.

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Factoring by Grouping

• Factor by grouping
• 2 groups of 2 terms
• Factor out the GCF
• from each group of 2 terms
• Factor out the
• common binomial factor

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Factoring by Grouping

• Factor
• by
• grouping

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Factoring Trinomialswith a coefficient of 1 for
the squared term

• Factor
• List the factors of 20
• Select the pairs from which 12
• may be obtained
• Write the two
• binomial factors
• Check using FOIL

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Factoring Trinomials ?TIP?

• ? If the last term of the trinomial is positive
  and the middle sign is positive, both binomials
  will have the same middle sign as the second
  term.

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Factoring Trinomials ?TIP?

• ? If the last term of the trinomial is positive
  and the middle sign is negative, both binomials
  will have the same middle sign as the second
  term.

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Factoring Trinomialswith a coefficient of 1 for
the squared term

• Factor
• List the factors of 22
• Select the pair from
• which 9 may be obtained
• Write the two
• binomial factors
• Check using FOIL

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Factoring Trinomials ?TIP?

• ? If the last term of the trinomial is negative,
  both binomials will have one plus and one minus
  middle sign.

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Factoring Trinomialsprimes

• A PRIME POLYNOMIAL cannot be factored using only
  integer factors.
• Factor
• The factors of 5 1 and 5.
• Since 2 cannot be obtained from 1 and 5, the
  polynomial is prime.

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Factoring Trinomials2 variables

• Factor
• The factors of 8 are 1,8 2,4, -1,-8 -2,
  -4
• Choose the pairs from which
• 6 can be obtained 2 4
• Use y in the first
• position and z in the
• second position
• Write the two binomial
• factors and
• check your answer

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Factoring Trinomialswith a GCF

• If there is a greatest common factor?
• If yes, factor it out first.

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Factoring Trinomialsalways check your factored
form

• Always check your answer with multiplication of
  the factors.

• The check

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Factoring Trinomialswhen the coefficient is not
1 on the squared term

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Factoring Trinomials---use grouping

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Factoring Trinomials---use grouping

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Factoring Trinomials---use FOIL and Trial and
Error

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Factoring Trinomials---use FOIL and Trial and
Error

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Factoring Trinomials---use FOIL and Trial and
Error

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Factoring Trinomials---use FOIL and Trial and
Error

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Factoring Trinomials---use FOIL and Trial and
Error

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Factoring Trinomials---use FOIL and Trial and
Error

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Factoring Trinomials---with a negative GCF

• Is the squared term negative?
• If yes, factor our a negative GCF.

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Special Factoringdifference of 2 squares

• The following must be true
• There must be only two terms in the polynomial.
• Both terms must be perfect squares.
• There must be a minus sign between the two
  terms.

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Special Factoringdifference of 2 squares

• The following pattern holds true for the
  difference of 2 squares

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Special Factoringdifference of 2 squares

• The pattern

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Special Factoringdifference of 2 squares

• The pattern

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Special Factoringdifference of 2 squares

• The pattern

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Special Factoringdifference of 2 squares

• The pattern

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Special Factoringperfect square trinomial

• A perfect square trinomial is a trinomial that is
  the square of a binomial.

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Special Factoringperfect square trinomial

• The first and third terms are perfect squares.
• AND the middle term is twice the product of the
  square roots of the first and third terms
• TEST THE MIDDLE TERM

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Special Factoringperfect square trinomial

• The patterns for a perfect square trinomial are

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Special Factoringperfect square trinomial

• Factor the following using the perfect square
  trinomial pattern

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Special Factoringperfect square trinomial

• Factor the following using the perfect square
  trinomial pattern

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Special Factoringdifference of two cubes

• Factor using the pattern.

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Special Factoringsum of two cubes

• Factor using the pattern.

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Solving quadratic equation with factoring

• A quadratic equation has a squared term.

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ZERO FACTOR PROPERTY

• To Factor a Quadratic,
• Apply the Zero-Factor Property.
• If a and b are real numbers and if ab 0, then a
  0 or b 0.

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Solving quadratic equations with
factoringZero-Factor Property

• Solve the equation
• (x 2)(x - 8) 0.
• Apply the zero-factor property
• (x 2) 0 or (x 8) 0
• x -2 or x 8

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Solving quadratic equations with
factoringZero-Factor Property

• There are two answers for x
• -2 and 8.
• Check by substituting the values calculated for x
  into the original equation.
• (x 2)(x - 8) 0.
• (-2 2)(-2 8) 0 (8 2)(8 8) 0
• 0 0
  0 0

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Solving quadratic equations with
factoringStandard Form

• To solve a quadratic equation,
• Write the equation in standard form.
• (Solve the equation for 0.)

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Solving quadratic equations with factoring

• To solve a quadratic equation,
• Factor the quadratic expression.

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Solving quadratic equations with factoring

• To solve a quadratic equation,
• Apply the Zero-Factor Property

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Solving quadratic equations with factoring

• To solve a quadratic equation,
• Check your answers