5-Minute Check on Activity 4-4

1. 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.

2. 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 a∙b = 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 book’s 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

6. Activity - Generalized In general, neglecting air resistance, an object’s 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

7. Activity - Analyzed The book’s 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

8. y x 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

9. 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

10. 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

11. 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)

12. Factoring Trinomials - Examples • x2 – 7x + 12 • x2 – 8x – 9 • 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)

13. Activity - Revisited The book’s 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

14. 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