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## Unit 6 – Chapter 9

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**Unit 6**• Chapter 8 Review and Chap. 8 Skills • Section 9.1 – Adding and Subtracting Polynomials • Section 9.2 – Multiply Polynomials • Section 9.3 – Special Products of Polynomials • Section 9.4 – Solve Polynomial Equations • Section 9.5 – Factor x2 + bx + c • Section 9.6 - Factor ax2 + bx + c • Section 9.7 and 9.8 – Factoring Special Products and Factoring Polynomials Completely**Vocabulary – X.X**• Holder • Holder 2 • Holder 3 • Holder 4**Notes – X.X – LESSON TITLE.**• Holder • Holder • Holder • Holder • Holder**–3 x**ANSWER 4x + 7 9x – 4 ANSWER ANSWER –7x – 4 ANSWER Prerequisite Skills SKILL CHECK Simplify the expression. 7. 3x +(– 6x) 8. 5 + 4x + 2 9. 4(2x – 1) +x 10. – (x + 4) – 6 x**27x3y3**x2y5 x15 –x3 ANSWER ANSWER ANSWER ANSWER 12.xy2xy3 Prerequisite Skills SKILL CHECK Simplify the expression. 11. (3xy)3 13. (x5)3 14. (– x)3**Vocabulary – 9.1**• Degree of a polynomial • Term with highest degree • Leading Coefficient • Coefficient of the highest degree term • Binomial • Polynomial with 2 terms • Trinomial • Polynomial with 3 terms • Monomial • Number, variable, or product of them • Degree of a Monomial • Sum of the exponents of the variables in a term • Polynomial • Monomial or Sum of monomials with multiple terms**Notes – 9.1 - Polynomials**• CLASSIFYING POLYNOMIALS • What is NOT a polynomial? Terms with: • Negative exponents • Fractional exponents • Variables as exponents • EVERYTHING ELSE IS A POLYNOMIAL! • To find DEGREE of a term • Add the exponents of each variable • Mathlish Polynomial Grammar • All polynomials are written so that the degree of the exponents decreases (i.e. biggest first)**Notes – 9.1 – Polynomials – Cont.**• I can only combine things in math that ……???? • ADDING POLYNOMIALS • Combine like terms • Remember to include the signs of the coefficients! • SUBTRACTING POLYNOMIALS • Use distributive property first!! • Combine like terms • Remember to include the signs of the coefficients!**EXAMPLE 1**Rewrite a polynomial Write 15x – x3 + 3 so that the exponents decrease from left to right. Identify the degree and leading coefficient of the polynomial. SOLUTION Consider the degree of each of the polynomial’s terms. 15x – x3 + 3 The polynomial can be written as – x3 +15x + 3. The greatest degree is 3, so the degree of the polynomial is 3, and the leading coefficient is –1.**Expression**Is it a polynomial? Classify by degree and number of terms a. 9 Yes 0 degree monomial b. 2x2 + x – 5 Yes 2nd degree trinomial c. 6n4 – 8n No; variable exponent d. n– 2 – 3 No; variable exponent e. 7bc3 + 4b4c Yes 5th degree binomial EXAMPLE 2 Identify and classify polynomials Tell whetheris a polynomial. If it is a polynomial, find its degree and classify it by the number of its terms. Otherwise, tell why it is not a polynomial.**EXAMPLE 3**Add polynomials Find the sum. a. (2x3 – 5x2 + x) + (2x2 + x3 – 1) b. (3x2 + x – 6) + (x2 + 4x + 10)**+ x3 + 2x2 – 1**EXAMPLE 3 Add polynomials SOLUTION a. Vertical format: Align like terms in vertical columns. (2x3 – 5x2 + x) 3x3 – 3x2 + x – 1 b. Horizontal format: Group like terms and simplify. (3x2 + x – 6) + (x2 + 4x + 10) = (3x2+ x2) + (x+ 4x) + (– 6+ 10) = 4x2 + 5x + 4**1.**Write 5y – 2y2 + 9 so that the exponents decrease from left to right. Identify the degree and leading coefficient of the polynomial. Degree is 2 Degree is 1 Degree is 0 EXAMPLE 1 for Examples 1,2, and 3 Rewrite a polynomial GUIDED PRACTICE SOLUTION Consider the degree of each of the polynomial’s terms. 5y –2y2 + 9 The polynomial can be written as – 2y2 +5y + 9. The greatest degree is 2, so the degree of the polynomial is 2, and the leading coefficient is –2**2.**Tell whether y3 – 4y + 3 is a polynomial. If it is a polynomial, find its degree and classify it by the number of its terms. Otherwise, tell why it is not a polynomial. EXAMPLE 2 for Example for Examples 1,2, and 3 Identify and classify polynomials GUIDED PRACTICE SOLUTION y3 – 4y + 3 is a polynomial.3 degree trinomial.**3.**Find the sum. EXAMPLE 3 for Example for Examples 1,2, and 3 Add polynomials GUIDED PRACTICE a. (2x3 + 4x – x) + (4x2 +3x3 – 6)**+ 3x3 + 4x2 – 6**(5x3+ 3x3) + (4x2) + (4x–2x) + (–6) EXAMPLE 3 for Example for Examples 1,2, and 3 Add polynomials GUIDED PRACTICE SOLUTION a. Vertical format: Align like terms in vertical columns. (5x3 + 4x – 2x) 8x3 + 4x2 +2x – 6 b. Horizontal format: Group like terms and simplify. (5x3 +4x – 2) + (4x2 + 3x3– 6) = = 8x3 + 4x2+ 2x – 6**EXAMPLE 4**Subtract polynomials Find the difference. a. (4n2 + 5) – (– 2n2 + 2n – 4) b. (4x2 – 3x + 5) – (3x2 – x – 8)**– (– 2n2 + 2n – 4)**2n2 – 2n + 4 EXAMPLE 4 Subtract polynomials SOLUTION a. (4n2 + 5) 4n2 + 5 6n2 – 2n + 9 b. (4x2 – 3x + 5) – (3x2 – x – 8) = 4x2 – 3x + 5– 3x2 + x + 8 = (4x2– 3x2) +(– 3x+x) + (5+ 8) =x2–2x+13**4.**Find the difference. EXAMPLE 4 for Examples 4 and 5 Subtract polynomials GUIDED PRACTICE a. (4x2 + 7x) – ( 5x2 + 4x – 9) (4x2 – 7x ) – (5x2 – 4x – 9) = 4x2 – 7x – 5x2 + 4x + 9 = (4x2– 5x2) +(– 7x– 4x) + 9 =–x2–11x +9**BASEBALL ATTENDNCE Look back at Example 5. Find**The difference in attendance at National and American League baseball games in 2001. 5. – 488t2 + 5430t + 24,700+ 318t2 – 3040t – 25,600 = = 7320 EXAMPLE 5 Solve a multi-step problem M =(– 488t2 + 5430t + 24,700) –(– 318t2 + 3040t + 25,600) = (– 488t2+ 318t2) + (5430t– 3040t) + (24,700 – 25,600) = – 170t2 + 2390t – 900 Substitute 6 for tin the model, because 2001 is 6 years after 1995.**ANSWER**About 7,320,000 people attended Major League Baseball games in 2001. EXAMPLE 5 Solve a multi-step problem**–18a + 2b**ANSWER 2.Simplifyr2s rs3. r3s4 6x2 + 4x x3 + 7x2 + 8x + 5 ANSWER ANSWER ANSWER Lesson 9.2, For use with pages 561-568 1.Simplify –2 (9a – b). 4.Simplifyx2(x+1) + 2x(3x+3) + 2x +5 3.Simplify 2x(3x + 2)**5200**ANSWER x2 + 3x +2 ANSWER Lesson 9.2, For use with pages 561-568 3. The number of hardback h and paperback p books (in hundreds) sold from 1999–2005 can be modeled by h = 0.2t2 – 1.7t + 14 and p = 0.17t3 – 2.7t2 + 11.7t + 27 where t is the number of years since 1999. About how many books sold in 2003. 4.Simplify (x + 1)(x + 2)**Vocabulary – 9.2**• Polynomial • Monomial or Sum of monomials with multiple terms**Notes – 9.2 – Multiply Polynomials**• Multiplying Polynomials is like using the distributive property over and over and over again. • Everything must be multiplied by everything else and combine like terms!!! • Frequently people use the FOIL process to multiply polynomials. • F – Multiply the First Terms • O – Multiply the Outside Terms • I – Multiply the Inside Terms • L – Multiply the Last Terms**EXAMPLE 1**Multiply a monomial and a polynomial Find the product 2x3(x3 + 3x2 – 2x + 5). 2x3(x3 + 3x2 – 2x + 5) Write product. = 2x3(x3) + 2x3(3x2) – 2x3(2x) + 2x3(5) Distributive property = 2x6 + 6x5 – 4x4 + 10x3 Product of powers property**EXAMPLE 2**Multiply polynomials using a table Find the product (x – 4)(3x + 2). SOLUTION STEP 1 Write subtraction as addition in each polynomial. (x – 4)(3x + 2) = [x + (– 4)](3x + 2)**ANSWER**The product is 3x2 + 2x – 12x – 8, or 3x2 – 10x – 8. 2 2 3x 3x x x 3x2 2x 3x2 – 12x – 8 – 4 – 4 EXAMPLE 2 Multiply polynomials using a table STEP2 Make a table of products.**x(7x2 +4)**1 for Examples 1 and 2 GUIDED PRACTICE Find the product. SOLUTION x(7x2 +4) Write product. = x(7x2 )+x(4) Distributive property =7x3+4x Product of powers property**ANSWER**(a +3)(2a +1) 2 The product is 2a2 + a + 6a + 3, or 2a2 + 7a + 3. 1 2a a 2a2 a 6a 3 3 for Examples 1 and 2 GUIDED PRACTICE Find the product. SOLUTION Make a table of products.**(4n – 1) (n +5)**3 for Examples 1 and 2 GUIDED PRACTICE Find the product. SOLUTION STEP 1 Write subtraction as addition in each polynomial. (4n – 1) (n +5) = [4n + (– 1)](n +5)**ANSWER**The product is 4n2 + 20n – n – 5, or 4n2 + 19n – 5. 5 n 4n 4n2 20n –n – 5 –1 for Examples 1 and 2 GUIDED PRACTICE STEP2 Make a table of products.**EXAMPLE 3**Multiply polynomials vertically Find the product (b2 + 6b – 7)(3b – 4). SOLUTION 3b3 + 14b2 – 45b + 28**Last**Outer Inner First EXAMPLE 4 Multiply polynomials horizontally Find the product (2x2 + 5x – 1)(4x – 3). Solution: Multiply everything and get = (2x2)(4x) + (2x2)(-3) + (5x)(4x) + (5x)(-3) + (-1)(4x) + (-1)(-3) = 8x3 + 14x2 – 19x + 3 FOIL PATTERN The letters of the word FOIL can help you to remember how to use the distributive property to multiply binomials. The letters should remind you of the words First, Outer, Inner, and Last. (2x + 3)(4x + 1) = 8x2 + 2x + 12x + 3**EXAMPLE 5**Multiply binomials using the FOIL pattern Find the product (3a + 4)(a – 2). (3a + 4)(a – 2) = (3a)(a) + (3a)(– 2) + (4)(a) + (4)(– 2) Write products of terms. = 3a2 + (– 6a) + 4a + (– 8) Multiply. = 3a2 – 2a – 8 Combine like terms.**4**(x2 + 2x +1)(x + 2) for Examples 3, 4, and 5 GUIDED PRACTICE Find the product. SOLUTION x3 + 4x2 + 5x + 2**(3y2 –y + 5)(2y – 3)**5 for Examples 3, 4, and 5 GUIDED PRACTICE Find the product. SOLUTION (3y2 –y + 5)(2y – 3) Write product. = 3y2(2y – 3) –y(2y – 3) + 5(2y – 3) Distributive property = 6y3 – 9y2 – 2y2 + 3y + 10y – 15 Distributive property = 6y3 – 11y2 + 13y – 15 Combine like terms.**(4b –5)(b – 2)**6 for Examples 3, 4, and 5 GUIDED PRACTICE Find the product. SOLUTION = (4b)(b) + (4b)(– 2) + (–5)(b) + (–5)(– 2) Write products of terms. = 4b2 – 8b – 5b + 10 Multiply. = 4b2 – 13b + 10 Combine like terms.**x2 + 5x + 6**x2 + 6x + 6 x2 + 6x x2 + 6 C B D A Area = length width EXAMPLE 6 Standardized Test Practice The dimensions of a rectangle are x + 3 and x + 2. Which expression represents the area of the rectangle? SOLUTION Formula for area of a rectangle = (x + 3)(x + 2) Substitute for length and width. =x2 + 2x + 3x + 6 Multiply binomials.**ANSWER**The correct answer is B. B A D C CHECK You can use a graph to check your answer. Use a graphing calculator to display the graphs of y1=(x + 3)(x + 2) and y2=x2 + 5x + 6 in the same viewing window. Because the graphs coincide, you know that the product of x + 3 and x + 2 is x2 + 5x + 6. EXAMPLE 6 Standardized Test Practice = x2 + 5x + 6 Combine like terms.**x2 + 14x + 49**ANSWER x2 - 14x + 49 x2– 49 9x2 + 3x – 2 ANSWER ANSWER ANSWER Lesson 9.3, For use with pages 569-574 Find the product. 1. (x + 7)(x + 7) 3. (x + 7)(x - 7) 2. (x - 7)(x - 7) 2. (3x – 1)(3x + 2)**15x2 + 46x + 16; 1344 m2**ANSWER Lesson 9.3, For use with pages 569-574 Find the product. 3. The dimensions of a rectangular playground can be represented by 3x + 8 and 5x + 2. Write a polynomial that represents the area of the playground. What is the area of the playground if x is 8 meters?