5-Minute Check on Activity 4-4

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5-Minute Check on Activity 4-4

• Using your calculator determine the solutions to
  the following
• y x2 6x 8
• y x2 2x 24
• y 4x2 8x
• -20 x2 2x 35
• 24 x2 5x

x 2 or x 4
x -4 or x 6
x 0 or x 2
x -3 or x 5
x -3 or x 8
Click the mouse button or press the Space Bar to
display the answers.
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Activity 4 - 5

• Sir Isaac Newton

3
Objectives

• Factor expressions by removing the greatest
  common factor
• Factor trinomials using trial and error
• Use the zero-product principle to solve equations
• Solve quadratic equations by factoring

4
Vocabulary

• Zero-product principle if ab 0, then either
  a 0 or b 0 or both equal 0.
• Factoring rewriting an expression as a product
  of two or more terms
• Common factor a factor that is multiplied in
  both terms
• Greatest common factor GCF, the largest common
  factor(s)

5
Activity

• Sir Isaac Newton XIV, a descendant of the famous
  physicist and mathematician, takes you to the top
  of a building to demonstrate a physics property
  discovered by his famous ancestor. He throws
  your math book straight up into the air. The
  books distance, s, above the ground as a
  function of time, x, is modeled by
• s(x) -16x2 16x 32
• When the book strikes the ground, what is the
  value of s?
• Write the equation you must solve to determine
  when.

s 0 feet
s 0 feet -16x2 16x 32
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Activity - Generalized

• In general, neglecting air resistance, an
  objects distance, s, above the ground as a
  function of time, t, is modeled by
• s(t) ½at2 v0t s0
• where t is the time
• a is the acceleration due to gravity
• 32 feet per second2 or
• 9.81 meters per second2
• v0 is the initial velocity that
  something is thrown at, measured in distance
  (feet or meters) per second
• s0 is the initial offset distance, how
  far above the ground did the object start at time
  t 0

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Activity - Analyzed

• The books distance, s, above the ground as a
  function of time, x, is modeled by
• s(x) -16x2 16x 32
• How tall is the building you were on top of?
• How fast did Newton throw the book up into the
  air?

s0 32 feet
v0 16 feet / sec
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Zero-Product Principle

• If a and b are any numbers and a?b 0, then
  either a 0, b 0, or both
• Example x(x 5) 0
• Example (x 2)(x 4) 0

Graph x2 5x 0
so either x 0 or x 5 0
so either x 0 or x 5
so either x 2 0 or x 4 0
so either x -2 or x 4
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Common Factors

• Common factor is a number or an expression that
  is a factor of each term of the entire expression
• Factoring is breaking the expression down into
  smaller parts multiplied together ? reverse
  distributive property
• Examples 3x 6 0 2x2
  8x 0

3 is in each term 3 (x 2) 0
2x is in each term 2x (x 4) 0
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Factoring Trinomials

• Reversing the FOIL method
• Break each term of trinomial down into its prime
  factors
• Remove the greatest common factor, GCF
• To factor the resulting trinomial into the
  product of two binomials, try combinations of
  factors for the first and last terms in two
  binomials
• Check the sum of the outer and inner products to
  match the middle term of the original trinomial
• If the constant term, c, is positive, both of its
  factors are positive or both are negative
• If the constant term is negative, one factor is
  positive and one is negative
• If the check fails, repeat steps 3 and 4

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Factoring Trinomials - Example

• Factor 4x3 8x2 32x

1) 2 2 x x x 2 2 2 x x 2
2 2 2 2 x
2) 4x (x2 2x 8)
3) x2 ? 1 1 8 ? ? 1 8 or 2
4
4) Since middle is -2 then we are looking for
one factor that is negative and 2 more than the
other ? -4 and 2
5) (x 4) ( x 2) x2 2x 4x 8 x2
2x 8 ?
4x3 8x2 32x factors into 4x(x 4)(x 2)
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Factoring Trinomials - Examples

1. x2 7x 12
2. x2 8x 9
3. x2 14x 49

x2 ? 1 1 12 ? 1 12, 2 6, or 3 4 both
minus (x 3) (x 4)
x2 ? 1 1 9 ? 1 9, or 3 3 one plus,
other minus (x 9) (x 1)
x2 ? 1 1 49 ? 1 49, or 7 7 both plus
(x 7) (x 7)
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Activity - Revisited

• The books distance, s, above the ground as a
  function of time, x, is modeled by
• s(x) -16x2 16x 32
• Solve the equation above by factoring
• At what time does the book hit the ground?

-16(x 2) (x 1)
-16(x 2) (x 1) 0 x 2 or x -1
negative value has no meaning book hits
ground 2 seconds after its thrown
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Summary and Homework

• Summary
• Factoring involves undoing the distributive
  property and breaking down into smaller products
• Factoring trinomials undoes the FOIL method
• Break each term of trinomial down into its prime
  factors
• Remove the greatest common factor, GCF
• To factor the resulting trinomial into the
  product of two binomials, try combinations of
  factors for the first and last terms in two
  binomials
• Check the sum of the outer and inner products to
  match the middle term of the original trinomial
• If the constant term, c, is positive, both of its
  factors are positive or both are negative
• If the constant term is negative, one factor is
  positive and one is negative
• If the check fails, repeat steps 3 and 4
• Solve quadratic equations by factoring
• Homework
• pg 445 446 problems 1, 2, 5, 8, 10, 14