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## ~ Chapter 9 ~

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**Algebra I**Algebra I Lesson 9-1 Adding & Subtracting Polynomials Lesson 9-2 Mulitplying and Factoring Lesson 9-3 Multiplying Binomials Lesson 9-4 Multiplying Special Cases Lesson 9-5 Factoring Trinomials of the Type x2 + bx + c Lesson 9-6 Factoring Trinomials of the Type ax2 + bx + c Lesson 9-7 Factoring Special Cases Lesson 9-8 Factoring by Grouping Chapter Review ~ Chapter 9 ~ Polynomials and Factoring**Lesson 9-1**Adding & Subtracting Polynomials Cumulative Review Chap 1-8**Lesson 9-1**Monomial – an expression that is a number, variable, or a product of a number and one or more variables. (Ex. 8, b, -4mn2, t/3…) (m/n is not a monomial because there is a variable in the denominator) Degree of a Monomial ¾ y Degree: 1 ¾ y = ¾ y1… the exponent is 1. 3x4y2 Degree: 6 The exponents are 4 and 2. Their sum is 6. -8 Degree: 0 The degree of a nonzero constant is 0. 5x0 Degree = ? Polynomial – a monomial or the sum or difference of two or more monomials. Standard form of a Polynomial… Simply means that the degrees of the polynomial terms decrease from left to right. 5x4 + 3x2 – 6x + 3 Degree of each? The degree of a polynomial is the same as the degree of the monomial with the greatest exponent. What is the degree of the polynomial above? Adding & Subtracting Polynomials Notes**Lesson 9-1**• 3x2 + 2x + 1 12 9x4 + 11x 5x5 • The number of terms in a polynomial can be used to name the polynomial. • Classifying Polynomials • Write the polynomial in standard form. • Name the polynomial based on its degree • Name the polynomial based on the number of terms • 6x2 + 7 – 9x4 3y – 4 – y3 8 + 7v – 11v • Adding Polynomials • There are two methods for adding (& subtracting) polynomials… • Method 1 – Add vertically by lining up the like terms and adding the coefficients. • Method 2 – Add horizontally by grouping like terms and then adding the coefficients. • (12m2 + 4) + (8m2 + 5) = Adding & Subtracting Polynomials Notes**Lesson 9-1**(9w3 + 8w2) + (7w3 + 4) = Subtracting Polynomials There are two methods for subtracting polynomials… Method 1 – Subtract vertically by lining up the like terms and adding the opposite of each term in the polynomial being subtracted. Method 2 – Subtract horizontally by writing the opposite of each term in the polynomial being subtracted and then grouping like terms. (12m2 + 4) - (8m2 + 5) = (30d3 – 29d2 – 3d) – (2d3 + d2) Adding & Subtracting Polynomials Notes**Lesson 9-1**Homework – Practice 9-1 Adding & Subtracting Polynomials Homework**Lesson 9-2**Multiplying & Factoring Practice 9-1**Lesson 9-2**Multiplying & Factoring Practice 9-1**Lesson 9-2**Multiplying & Factoring Practice 9-1**Lesson 9-2**Distributing a monomial 3x(2x - 3) = 3x(2x) – 3x(3) = -2s(5s - 8) = -2s(5s) – (-2s) (8) = Multiplying a Monomial and a Trinomial 4b(5b2 + b + 6) = 4b(5b2) + 4b(b) + 4b(6) = -7h(3h2 – 8h – 1) = 2x(x2 – 6x + 5) = Factoring a Monomial from a Polynomial Find the GCF for 4x3 + 12x2 – 8x 4x3 = 2*2*x*x*x 12x2 = 2*2*3*x*x 8x = 2*2*2*x What do they all have in common? Mulitplying & Factoring Notes 2*2*x = 4x**Lesson 9-2**Find the GCF of the terms of 5v5 + 10v3 Find the GCF of the terms of 4b3 – 2b2 – 6b Factoring out a Monomial Step 1: Find the GCF Step 2: Factor out the GCF… Factor 8x2 – 12x = Factor 5d3 + 10d = Factor 6m3 – 12m2 – 24m = Factor 6p6 + 24p5 + 18p3 = Multiplying & Factoring Notes**Lesson 9-2**Homework ~ Practice 9-2 even Multiplying & Factoring Homework**Lesson 9-3**Multiplying Binomials Practice 9-2**Lesson 9-3**Using the Distributive Property Simplify (6h – 7)(2h + 3) = 6h(2h + 3) – 7(2h + 3) = (5m + 2)(8m – 1) = 5m(8m – 1) + 2(8m - 1) = (9a – 8)(7a + 4) = 9a(7a + 4) – 8(7a + 4) = Multiplying using FOIL F = First O = Outer I = Inner L = Last (6h – 7)(2h + 3) = 6h(2h) + 6h(3) + (-7)(2h) + (-7)(3) 12h2 + 18h + (-14h) + (-21) = 12h2 + 4h -21 (3x + 4)(2x + 5) = (3x – 4)(2x – 5) = Applying Multiplication of Polynomials Determine the area of each rectangle and subtract the area of center (x + 8)(x + 6) = 3x(x + 3) = Multiplying Binomials Notes**Lesson 9-3**Multiplying a Trinomial and a Binomial (2x – 3)(4x2 + x -6) = 2x(4x2) + 2x(x) + 2x(-6) -3(4x2) -3(x) -3(-6) 8x3 + 2x2 + (-12x) - 12x2 -3x + 18 Combine like terms = 8x3 – 10x2 – 15x + 18 You can also multiply using the vertical multiplication method… Try this one… (6n – 8)(2n2 + n + 7) = Multiplying Binomials Notes**Lesson 9-3**Homework – Practice 9-3 even Multiplying Binomials Homework**Lesson 9-4**Multiplying Special Cases Practice 9-3**Lesson 9-4**Multiplying Special Cases Practice 9-3**Lesson 9-4**Finding the Square of a Binomial (x + 8)2 = (x + 8)(x + 8) = So… (a + b)2 = Rule: The Square of a Binomial (a + b)2 = a2+ 2ab + b2 (a – b)2 = a2 – 2ab + b2 Find (t + 6)2 (5y + 1)2 (7m – 2p)2 Find the Area of the shaded region… (x + 4)2 – (x – 1)2 Mental Math – Squares 312 = (30 + 1)2 = 302 + 2(30*1) + 12 = 900 + 60 + 1 = 961 Multiplying Special Cases Notes**Lesson 9-4**292 = 982 = Difference of Squares (a + b)(a – b) = a2 – ab + ab – b2 = a2 – b2 Find each product. (d + 11)(d – 11) = d2 – 112 = d2 – 121 (c2 + 8)(c2 – 8) = (9v3 + w4)(9v3 – w4) = Mental Math 18 * 22 = (20 + 2)(20 – 2) = 202 – 22 = 400 – 4 = 396 59 * 61 = 87 * 93 = Multiplying Special Cases Notes**Lesson 9-4**Homework – Practice 9-4 odd Multiplying Special Cases Homework**Lesson 9-5**Factoring Trinomials of the Type x2 + bx + c Practice 9-4**Lesson 9-5**Factoring Trinomials of the Type x2 + bx + c Practice 9-4**Lesson 9-5**Factoring Trinomials x2 + bx + c To factor this type of trinomial… you must find two numbers that have a sum of b and a product of c. Factor x2 + 7x + 12 Make a table… Column 1 lists factors of c 12… Column 2 lists the sum of those factors… b Row 3 – factors 3 & 4 with a sum of 7 fits so… x2 + 7x + 12 = (x + 3)(x + 4) Factor g2 + 7g + 10 Factor a2 + 13a + 30 Factoring Trinomials of the Type x2 + bx + c Notes**Lesson 9-5**Factoring x2 – bx + c Since the middle term is negative, you must find the negative factors of c, whose sum is –b. Factor d2 – 17d + 42 > Make a table… Row 3 – factors -3 & -14 with sum of -17 So… d2 – 17d + 42 = (d – 3)(d – 14) Factor k2 – 10k + 25 Factor q2 – 15q + 36 Factoring Trinomials with a negative c (- c) Factor m2 + 6m - 27 Make a table Row 4 – factors 9 & -3 with sum of 6 Factoring Trinomials of the Type x2 + bx + c Notes**Lesson 9-5**So… m2 + 6m – 27 = (m + 9)(m – 3) Factor p2 – 3p – 40 Factor m2 + 8m – 20 Factor y2 – y - 56 Factoring Trinomials of the Type x2 + bx + c Notes**Lesson 9-5**Homework ~ Practice 9-5 #1-30 Factoring Trinomials of the Type x2 + bx + c Homework**Lesson 9-6**Factoring Trinomials of the Type ax2 + bx + c Practice 9-5**Lesson 9-6**Factoring Trinomials of the Type ax2 + bx + c Practice 9-5**Lesson 9-6**Factoring Trinomials of the Type ax2 + bx + c Practice 9-5**Lesson 9-6**Factoring Trinomials when c is positive 6n2 + 23n + 7… Multiply a & c So… 6n2 + 2n + 21n + 7 Factor using GCF 2n(3n + 1) + 7(3n + 1) (2n + 7)(3n + 1) = 6n2 + 23n + 7 Try another one… 2y2 + 9y + 7 So… 2y2 + 2y +7y + 7 Factor… 2y(y + 1) + 7(y + 1) (2y + 7)(y + 1) What if b is negative? 6n2 – 23n + 7 6n2 - 2n – 21n + 7 Factoring Trinomials of the Type ax2 + bx + c Notes**Lesson 9-6**2n(3n - 1) – 7(3n – 1) (2n – 7)(3n – 1) Your turn… 2y2 – 5y + 2 Factoring Trinomials when c is negative… 7x2 – 26x – 8 7x2 -28x + 2x – 8 7x(x – 4) + 2(x – 4) (7x + 2)(x – 4) Factor5d2 – 14d – 3 5d2 -15d + 1d – 3 5d(d – 3) + 1(d – 3) (5d + 1)(d - 3) Factoring Trinomials of the Type ax2 + bx + c Notes**Lesson 9-6**Factoring Out a Monomial First 20x2 + 80x + 35 Factor out the GCF first… 5(4x2 + 16x + 7)… then factor 4x2 + 16x + 7 4x2 + 2x + 14x + 7 2x(2x + 1) + 7(2x + 1) (2x + 7)(2x + 1) Remember to include the GCF in the final answer 5(2x + 7)(2x + 1) Factor 18k2 – 12k - 6 6(3k2 – 2k – 1) 3k2 - 3k + 1k – 1 3k(k – 1) + 1(k – 1) = 6(3k + 1)(k - 1) Factoring Trinomials of the Type ax2 + bx + c Notes**Lesson 9-6**Homework: Practice 9-6 first column Factoring Trinomials of the Type ax2 + bx + c Homework**Lesson 9-7**Factoring Special Cases Practice 9-6**Lesson 9-7**Factoring Special Cases Practice 9-6**Lesson 9-7**Perfect Square Trinomials a2 + 2ab + b2 = (a + b)(a + b) = (a + b)2 a2 – 2ab + b2 = (a – b)(a – b) = (a – b)2 So… x2 + 12x + 36 = (x + 6)2 And… x2 – 14x + 49 = (x – 7)2 What about… 4x2 + 12x + 9 Factoring a Perfect-Square Trinomial with a = 1 (ax2 + bx + c) x2 + 8x + 16 = n2 – 16n + 64 = Factoring a Perfect-Square Trinomial with a ≠ 1 9g2 – 12g + 4 4t2 + 36t + 81 Factoring Special Cases Notes**Lesson 9-7**Factoring the Difference of Squares a2 – b2 = (a + b)(a – b) Or… x2 – 16 = What about 25x2– 81 = Try x2 – 36 Factor 4w2 – 49 Look for common factors… 10c2 – 40 = 28k2 – 7 = 3c4 – 75 = Factoring Special Cases Notes**Lesson 9-7**Homework: Practice 9-7 odd #1-39 Factoring Special Cases Homework**Lesson 9-8**Factoring by Grouping Practice 9-7**Lesson 9-8**Factoring by Grouping Practice 9-7**Lesson 9-8**Factoring by Grouping Practice 9-7**Lesson 9-8**Factoring a Four-Term Polynomial 4n3 + 8n2 – 5n – 10 Factor the GCF out of each group of 2 terms. ? (4n3 + 8n2) - ? (5n + 10) Factor 5t4 + 20t3 + 6t + 24 Before you factor, you may need to factor out the GCF. 12p4 + 10p3 -36p2 – 30p Try… 45m4 – 9m3 + 30m2 – 6m (factor completely) Finding the dimensions of a rectangular prism The volume (lwh) of a rectangular prism is 80x3 + 224x2 + 60x. Factor to find the possible expressions for the length, width, and height of the prism. Factoring by Grouping Notes**Lesson 9-8**Your turn… Find expressions for possible dimensions of the rectangular prism… V = 6g3 + 20g2 + 16g V = 3m3 + 10m2 + 3m Factoring by Grouping Notes**Lesson 9-8**Classwork – Practice 9-8 even # 1-28 Factoring by Grouping Homework**Lesson 9-8**Factoring by Grouping Practice 9-8**Chap 9 Quiz Review Lesson 7 & 8**Practice 9-8**Algebra I**Algebra I ~ Chapter 9 ~ Chapter Review**Algebra I**Algebra I ~ Chapter 9 ~ Chapter Review