Polynomials and Factoring Algebra I Algebra I ~ Chapter 9 ~ Polynomials and Factoring 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

Adding & Subtracting Polynomials Lesson 9-1 Cumulative Review Chap 1-8

Adding & Subtracting Polynomials Lesson 9-1 Notes 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 Lesson 9-1 Notes 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 Lesson 9-1 Notes (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 Lesson 9-1 Adding & Subtracting Polynomials Homework Homework – Practice 9-1

Multiplying & Factoring Lesson 9-2 Practice 9-1

Multiplying & Factoring Lesson 9-2 Practice 9-1

Multiplying & Factoring Lesson 9-2 Practice 9-1

Mulitplying & Factoring Lesson 9-2 Notes 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? 2*2*x = 4x

Multiplying & Factoring Lesson 9-2 Notes 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 Lesson 9-2 Multiplying & Factoring Homework Homework ~ Practice 9-2 even

Multiplying Binomials Lesson 9-3 Practice 9-2

Multiplying Binomials Lesson 9-3 Notes 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 Lesson 9-3 Notes 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 Lesson 9-3 Multiplying Binomials Homework Homework – Practice 9-3 even

Multiplying Special Cases Lesson 9-4 Multiplying Special Cases Practice 9-3

Multiplying Special Cases Lesson 9-4 Multiplying Special Cases Practice 9-3

Multiplying Special Cases Lesson 9-4 Notes 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 Lesson 9-4 Notes 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 Lesson 9-4 Homework Homework – Practice 9-4 odd

Factoring Trinomials of the Type x2 + bx + c Lesson 9-5 Practice 9-4

Factoring Trinomials of the Type x2 + bx + c Lesson 9-5 Practice 9-4

Factoring Trinomials of the Type x2 + bx + c Lesson 9-5 Notes 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 Lesson 9-5 Notes 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 Lesson 9-5 Notes 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 Lesson 9-5 Homework Homework ~ Practice 9-5 #1-30

Factoring Trinomials of the Type ax2 + bx + c Lesson 9-6 Factoring Trinomials of the Type ax2 + bx + c Practice 9-5

Factoring Trinomials of the Type ax2 + bx + c Lesson 9-6 Factoring Trinomials of the Type ax2 + bx + c Practice 9-5

Factoring Trinomials of the Type ax2 + bx + c Lesson 9-6 Practice 9-5

Factoring Trinomials of the Type ax2 + bx + c Lesson 9-6 Factoring Trinomials of the Type ax2 + bx + c Notes 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 Factors of a*c Sum (=b) 6 and 7 13 3 and 14 17 2 and 21 23 √

Factoring Trinomials of the Type ax2 + bx + c Lesson 9-6 Factoring Trinomials of the Type ax2 + bx + c Notes 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) Factor 5d2 – 14d – 3 5d2 -15d + 1d – 3 5d(d – 3) + 1(d – 3) (5d + 1)(d - 3) Factors of a*c Sum (=b) 7 and -8 -1 4 and -14 -10 2 and -28 -26 √

Factoring Trinomials of the Type ax2 + bx + c Lesson 9-6 Factoring Trinomials of the Type ax2 + bx + c Notes 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 Lesson 9-6 Factoring Trinomials of the Type ax2 + bx + c Homework Homework: Practice 9-6 first column

Factoring Special Cases Lesson 9-7 Practice 9-6

Factoring Special Cases Lesson 9-7 Practice 9-6

Factoring Special Cases Lesson 9-7 Notes 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 Lesson 9-7 Notes 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 Lesson 9-7 Homework Homework: Practice 9-7 odd #1-39

Factoring by Grouping Lesson 9-8 Practice 9-7

Factoring by Grouping Lesson 9-8 Practice 9-7

Factoring by Grouping Lesson 9-8 Practice 9-7

Factoring by Grouping Notes Lesson 9-8 4n3 + 8n2 – 5n – 10 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 V = 6g3 + 20g2 + 16g Your turn… Find expressions for possible dimensions of the rectangular prism… V = 6g3 + 20g2 + 16g V = 3m3 + 10m2 + 3m

Classwork – Practice 9-8 even Factoring by Grouping Lesson 9-8 Homework Classwork – Practice 9-8 even # 1-28

Factoring by Grouping Lesson 9-8 Practice 9-8

Chap 9 Quiz Review Lesson 7 & 8 Practice 9-8

~ Chapter 9 ~ Algebra I Algebra I Chapter Review

~ Chapter 9 ~ Algebra I Algebra I Chapter Review