Splash Screen Unit 8 Quadratic Expressions and Equations EQ: How do you use addition, subtraction, multiplication, and factoring of polynomials in order.

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Splash Screen Unit 8 Quadratic Expressions and Equations EQ: How do you use addition, subtraction, multiplication, and factoring of polynomials in order to simplify rational expressions?

Splash Screen EQ: How do you use the distributive property (GCF) to factor polynomials and solve equations of the form ax 2 + bx = 0? Lesson 5 Using the Distributive Property

Lesson Menu 5 minute check on previous lesson. Do the first 5 problems!

Over Lesson 8–4 5-Minute Check 1 A.16x B. 16x x + 25 C. 16x x + 25 D. 4x x + 5 Find (4x + 5) 2.

Over Lesson 8–4 5-Minute Check 2 A.15a 2 – 30ab + 15b 2 B. 9a 2 – 30ab + 25b 2 C. 9a 2 – 15ab + 25b 2 D. 3a 2 – 15ab + 5b 2 Find (3a – 5b) 2.

Over Lesson 8–4 5-Minute Check 3 A.9x x – 16 B.9x 2 – 24x – 16 C.9x D.9x 2 – 16 Find (3x + 4)(3x – 4).

Over Lesson 8–4 5-Minute Check 4 A.4c 2 – 36d 2 B.4c d 2 C.4c cd + 36d 2 D.4c cd – 36d 2 Find (2c 2 + 6d)(2c 2 – 6d).

Over Lesson 8–4 5-Minute Check 5 A.(x + 3) 2 (x – 6) 2 B.2x 2 – 6x + 45 C.(x + 3) 2 + (x – 6) 2 D.2x Write a polynomial that represents the area of the figure at the right.

Splash Screen EQ: How do you use the distributive property (GCF) to factor polynomials and solve equations of the form ax 2 + bx = 0? Lesson 5 Using the Distributive Property

Then/Now Used the Distributive Property to evaluate expressions. Use the Distributive Property to factor polynomials. Solve quadratic equations of the form ax 2 + bx = 0. EQ: How do you use the distributive property (GCF) to factor polynomials and solve equations of the form ax 2 + b = 0?

Then/Now Old Vocabulary Greatest Common Factor (GCF) – the largest factor that two or more numbers or algebraic terms have in common. Distributive Property a(b + c) = ab + ac ab + ac = a(b + c)

Then/Now

Vocabulary factoring - to express or write a number, monomial, or polynomial as a product of two or more factors. factoring by grouping – using the distributive property to factor polynomials with four or more terms; terms are put into groups, then factored. Zero Product Property

Example 1 Use the Distributive Property A. Use the Distributive Property to factor 15x + 25x 2. First, find the GCF of 15x + 25x 2. 15x = 3 ● 5 ● x Factor each monomial. Circle the common prime factors. GCF = 5 ● x or 5x Write each term as the product of the GCF and its remaining factors. Then use the Distributive Property to factor out the GCF. 25x 2 = 5 ● 5 ● x ● x = 5x (3 + 5x)Distributive Property 15x + 25x 2 = 5x(3) + 5x(5 ● x) Rewrite each term using GCF. Answer: 5x (3 + 5x).

Example 1 Use the Distributive Property B. Factor 12xy + 24xy 2 – 30x 2 y 4. 12xy=2 ● 2 ● 3 ● x ● y 24xy 2 =2 ● 2 ● 2 ● 3 ● x ● y ● y –30x 2 y 4 =–1 ● 2 ● 3 ● 5 ● x ● x ● y ● y ● y ● y GCF = 2 ● 3 ● x ● y or 6xy Circle common factors. Factor each term. = 6xy (2 + 4y – 5xy 3 ) 12xy + 24xy 2 – 30x 2 y 4 = 6xy(2) + 6xy(4y) + 6xy(–5xy 3 ) Answer: 6xy(2 + 4y – 5xy 3 )

Example 1 A.3xy (x + 4y) B.3 (x 2 y + 4xy 2 ) C.3x (xy + 4y 2 ) D.xy (3x + 2y) C. Factor 3x 2 y + 12xy 2.

Example 1 A.3(ab 2 + 5a 2 b 2 + 9ab 3 ) B.3ab(b + 5ab + 9b 2 ) C.ab(b + 5ab + 9b 2 ) D.3ab 2 (1 + 5a + 9b) D. Factor 3ab a 2 b ab 3.

Example 1 EXTRA 1. Factor 27y y.

Example 1 EXTRA 2. Factor 4a 2 b – 8ab 2 + 2ab.

End of the Lesson Assignment Do Worksheet #1 to #12 Assignment Do Worksheet #1 to #12 EQ: How do you use the distributive property (GCF) to factor polynomials and solve equations of the form ax 2 + bx = 0?

Concept

Example 2 Factor by Grouping A. Factor 2xy + 7x – 2y – 7. 2xy + 7x – 2y – 7 = (2xy – 2y) + (7x – 7)Group terms with common factors. = 2y(x – 1) + 7(x – 1)Factor the GCF from each group. = (x – 1)(2y + 7) Distributive Prop Answer: (x – 1)(2y + 7) or (2y + 7)(x – 1)

Example 2 Factor by Grouping EXTRA 1. Factor 4mn + 8n + 3m + 6.

Example 2 Factor by Grouping EXTRA 2. Factor xy + 5y – x – 5.

Example 2 Factor by Grouping EXTRA 3. Factor 3np + 15p – 4n – 20.

Example 3 Factor by Grouping EXTRA 4. Factor 3p – 2p 2 – 18p + 27.

Example 2 A.(4x – 5)(y + 3) B.(7x + 5)(2y – 3) C. (4x + 3)(y – 5) D. (4x – 3)(y + 5) B. Factor 4xy + 3y – 20x – 15.

Example 3 Factor by Grouping with Additive Inverses C. Factor 15a – 3ab + 4b – a – 3ab + 4b – 20 = (15a – 3ab) + (4b – 20)Group terms with common factors. = 3a(5 – b) + 4(b – 5)Factor the GCF from each group. = 3a(–1)(b – 5) + 4(b – 5)5 – b = –1(b – 5) = – 3a(b – 5) + 4(b – 5)3a(–1) = –3a = (b – 5)(– 3a + 4) Distributive Property Answer: (b – 5)(–3a + 4) or (b – 5)(4 – 3a)

Example 3 Factor by Grouping with Additive Inverses EXTRA 1. Factor 2mk – 12m + 42 – 7k.

Example 3 Factor by Grouping with Additive Inverses EXTRA 2. Factor c – 2cd + 8d – 4.

Example 3 A.(2x – 3)(y – 5) B.(–2x + 3)(y + 5) C.(3 + 2x)(5 + y) D.(–2x + 5)(y + 3) D. Factor – 2xy – 10x + 3y + 15.

End of the Lesson Assignment Do Worksheet #13 to #20 Assignment Do Worksheet #13 to #20 EQ: How do you use the distributive property (GCF) to factor polynomials and solve equations of the form ax 2 + bx = 0?

Concept If ab=0, then a=0, b=0, or both a=0 and b=0.

Example 4 Solve Equations A. Solve (x – 2)(4x – 1) = 0. Check the solution. If (x – 2)(4x – 1) = 0, then according to the Zero Product Property, either x – 2 = 0 or 4x – 1 = 0. (x – 2)(4x – 1) = 0Original equation x – 2 = 0 or 4x – 1= 0 Zero Product Property x = 2 4x= 1Solve each equation. Divide.

Example 4 Solve Equations (x – 2)(4x – 1)=0 (x – 2)(4x – 1) = 0 Check Substitute 2 and for x in the original equation.(2 – 2)(4 ● 2 – 1) = 0 ? ? (0)(7) = 0 ?? 0 = 0 0 = 0

Example 4 Solve Equations EXTRA 1. Solve (2x + 6)(3x – 15) = 0. Check.

Example 4 Solve Equations B. Solve 4y = 12y 2. Check the solution. Write the equation so that it is of the form ab = 0. 4y=12y 2 Original equation 4y – 12y 2 =0Subtract 12y 2 from each side. 4y(1 – 3y)=0Factor the GCF of 4y and 12y 2, which is 4y. 4y = 0 or 1 – 3y=0Zero Product Property y = 0–3y=–1Solve each equation. Divide. Answer: The roots are 0 and ⅓.

Solve Equations EXTRA 2. Solve x 2 = 3x. Check.

Solve Equations EXTRA 3. Solve 8x 2 – 40x = 0. Check.

Solve Equations EXTRA 4. Solve x 2 = - 10x. Check.

Example 4 A.{3, –2} B.{–3, 2} C.{0, 2} D.{3, 0} C. Solve (s – 3)(3s + 6) = 0. Check the solution.

Example 4 D. Solve 5x – 40x 2 = 0. Check the solution. A.{0, 8} B. C.{0} D.

Example 5 Use Factoring FOOTBALL A football is kicked into the air. The height of the football can be modeled by the equation h = –16x x, where h is the height reached by the ball after x seconds. Find the values of x when h = 0. h= –16x x Original equation 0= –16x x h = 0 0= 16x(–x + 3)Factor by using the GCF. 16x= 0 or –x + 3= 0Zero Product Property x= 0 3= xSolve each equation. Answer: At 0 seconds or 3 seconds the football is at a height of 0.

Example 5 A.0 or 1.5 seconds B.0 or 7 seconds C.0 or 2.66 seconds D.0 or 1.25 seconds Juanita is jumping on a trampoline in her back yard. Juanita’s jump can be modeled by the equation h = –14t t, where h is the height of the jump in feet at t seconds. Find the values of t when h = 0.

End of the Lesson Assignment Do Worksheet #21 to #40 Assignment Do Worksheet #21 to #40 EQ: How do you use the distributive property (GCF) to factor polynomials and solve equations of the form ax 2 + bx = 0?