Chapter 5A: Polynomials Algebra 2A -2012

Lesson 5.1A I can identify the base and the exponent. I can simplify product of monomials.

bn Power What is it? Examples: Important characteristic: Nonexamples: Base: ____ Exponent: ____ Shortcut for repeated multiplications

(45)(42) (6x2)(x4) (3x4y)(-4x2) Example 1: Simplify the following a. b. c.

(23)(24) (5v4)(3v) (-4ab6)(7a2b3) Your Turn 1: Simplify the following 1. 2. 3.

bn • bm = bn + m same bases Product of Powers x3 • x4 (2x3y)(5x) What is it? Examples: x3 • x4 bn • bm = bn + m same bases (2x3y)(5x) Product of Powers Important characteristic: Nonexamples: Add the exponents x3 • x4 ≠ x12

(x5)2 (y7)3 I can simplify product of Powers. Example 2: Simplify the following a. b. c. (3xy)2 (x5)2 (y7)3

Your Turn 2: Simplify the following (2b2)4 (4abc)2 1. 2. 3.

Example 3: Simplify the following (10ab4)3 (3b2)2 d.

Your Turn 3: Simplify the following (2xy2)3 (-4x5)2 4.

I can simplify quotient of monomials. Example 4 a. 16x3y4 4 x5y b.

Your Turn! (3xy5)2 (2x3y7)3

Homework: Lesson 5.1 A Check your answers !

Lesson 5.1B I can simplify quotient of monomials. I can simplify monomials with negative exponents. I can simplify monomials with a zero exponent.

(5xy)0 = 1 (5xy)0 ≠ 0 (b)0 = 1 Zero Exponents 8-2 Anything to the power of zero is one! (5xy)0 ≠ 0

(b)-n = (5a)-1 = (5a)-1 ≠ -5a Negative Exponents 8-2 “move” up or down to make it a positive exp. (5a)-1 ≠ -5a

Negative exponent rule: a-n = a-n is the reciprocal of an Example1: 1. x-3 2. 5a-2b0 3. -4a-7b-3

Your Turn 1: a. 3w-3 b. 4x-7 c. 3x0y-4

Example 2: 4. 5.

Your Turn 2: d. e. f.

Learning target(s): I can multiply and divide monomials. (LT1) Mixed practice: Simplify expressions means to write expressions without parentheses or negative exponents. a. b. c.

Learning target(s): I can multiply and divide monomials. (LT1) Mixed practice: d. e. f.

Learning targets: I can multiply and divide monomials. (LT1) Mixed practice: g. h.

Learning target(s): I can multiply and divide monomials. (LT1) Mixed practice: Simplify expressions means to write expressions without parentheses or negative exponents. a. b. c.

Learning target(s): I can multiply and divide monomials. (LT1) Mixed practice: d. e. f.

Learning targets: I can multiply and divide monomials. (LT1) Mixed practice: g. h.

Homework: Lesson 5.1B

Warm- up over Lesson 5.1A & B Simplify the following. 1. 2. 3. 4. 5. 6. Write a problem involving operations with powers whose answer is: -8x3

5.1B

Lesson 5.2 (LT3) I can recognize a monomial, binomial, or trinomial. I can identify the degree of a polynomial. I can rewrite polynomials in descending order. I can add polynomials. I can subtract polynomials. I can multiply polynomials.

Definition of monomial

Classify as monomial or non-monomial Example 1a: Examples: Non Examples:

Definition of Polynomials

Classify as polynomial or non-polynomial Example 1b: Examples: Non Examples: 51

a. 2x3 + 4x2 4 b. 2xy3 – 4x4y0 + 2x c. x2 + 3xy y d. 4x-2 Your Turn 1: Is it a polynomial? Yes or No? Classify as monomial, binomial, or trinomial? a. 2x3 + 4x2 4 b. 2xy3 – 4x4y0 + 2x c. x2 + 3xy y d. 4x-2

Descending: decreasing (biggest to smallest) Example 2: Arrange the terms of the polynomial so that the powers of x are in descending order Descending: decreasing (biggest to smallest) a. 4x2 + 7x3 + 5x b. 9x3y – 4x5 + 8y - 6xy4 Your turn 2: c. 10 + 7x3y – 2x4y2 + 8x

a. 5x2 b. -9x3y5 Your Turn 3: c. 7xy5z4 d. 10 Example 3: Find the degree of the monomial. Degree of a monomial: sum of the exponents of the variables a. 5x2 b. -9x3y5 Your Turn 3: c. 7xy5z4 d. 10

a. 5x4 + 3x2 – 9x Your Turn 4: b. 4x3 – 7x + 5x6 Example 4: Find the degree of the polynomial. Degree of a polynomial: it is the highest degree after finding the degree of each term. a. 5x4 + 3x2 – 9x Your Turn 4: b. 4x3 – 7x + 5x6

Check for Understanding Lesson 5.2 A: (LT3) Is the polynomial a monomial, binomial, or trinomial. a. y3 – 4 b. 3y3 + y0 – 2x c. -4xy5 Arrange the terms of the polynomial so that the powers of x are in descending order. d. 3x4y3 – x6y0 + 6 – 2x Find the degree of the monomial e. -2x4y3z f. 3ab0y3 Find the degree of the polynomial. g. 3x4y3 – x6y0 + 6 – 2x

(3x2 – 4x + 8) + (2x -7x2 -5) Example 5: Simplify the following 2. (5y2 – 3y + 8) + (4y2 – 9) Your Turn 5: Simplify the following (LT4) (3x2 – 4x + 8) + (2x -7x2 -5)

(3x2 – 4x + 8) – (2x -7x2 -5) 3. (2x2 – 3x + 1) – (x2 + 2x – 4) Example 6: Simplify the following 3. (2x2 – 3x + 1) – (x2 + 2x – 4) 4. (4y2 – 9) – (5y2 – 3y + 8) Your Turn 6: Simplify the following (LT4) (3x2 – 4x + 8) – (2x -7x2 -5)

Example 7: Simplify the following. 1. y( 7y + 9y2) 2. -3x(2x2 + xy – 3)

Your Turn 7: Simplify the following. 1. n2(3n2 + 7) 2. n2(3n2p - np + 7p)

Example 8: Multiply the following. 1. (x + 3)(x + 2) 2. (x  9)(3x + 7) 3.

Your Turn 8: Find the product. (LT5) a. b.

Simplify. c. d. e.

Homework: Practice 5.2

Mixed Review Lessons 5.1 & 5.2: Simplify the following 1. (-2a2 b0)3 = 2. 3x2y(-4x + 2y3) = 3. (2x2 – 3x – 10) – (3x + 4x2 – 4) Find the degree of the polynomial. 4. -52x2yz4 – 7x4y3z

Mixed Review Lessons 5.1 & 5.2 Simplify the following. 5. (3x + 2y) + (3x – 2y) 6. (3x + 2y) – (3x – 2y) 7. 4x(3x + 2y) 8. (3x + 2y)(3x – 2y)

Mixed Review Lessons 5.1 & 5.2 Simplify the following 9. (x3 y 2)(x 4 y6) = 10. (32x5 y8) 2 = 11. (4x 2 – 8x + 10) – (x 2 – 12) 12. 13. Find the degree of the polynomial. 14. 3x2y5 – 10xy8

Simplify the following. 15. 16. 17. 18. 19. 20. Write a problem involving operations with polynomials whose answer is: 9x2 – 30x + 25

Dividing Polynomials lesson 5-3 A Learning targets: I can divide polynomials by a monomial.(LT6) I can divide polynomials using long division. (LT7) Notes first! We will start the Quiz at

Expressions that represent division: Non-examples:

To divide a polynomial by a monomial, use the properties of powers from lesson 5-1. Example 1: Simplify.

Example 2: Simplify

Your Turn 1: Simplify

Review: How to divide using long division.

To divide by a polynomial by a polynomial, use a long division pattern. Remember that only like terms can be added or subtracted. Example 3: Use long division to find the following.

Example 4: Use long division to divide.

Example 5:

Your Turn 2: Divide using long division.

Your Turn 2: Divide using long division.

Your Turn 2: Divide using long division.

Closure Lesson 5.3a: Dividing Polynomials Homework: Practice 5.3 part I

Closure Lesson 5.3a: Dividing Polynomials Homework: Practice 5.3 part I

Simplify the following. 1. Warm- up Lesson 5.3a Simplify the following. 1. 2. 3. 4. 5. Write a problem involving division of polynomials whose answer is: x - 3

Part I

Dividing Polynomials lesson 5-3 B Learning targets: I can divide polynomials using synthetic division. (LT8)

Use Synthetic Division: A procedure to divide a polynomial by a binomial using coefficients of the dividend and the value of r in the divisor x – r. Example 6: Use synthetic division to divide.

Use Synthetic Division: A procedure to divide a polynomial by a binomial using coefficients of the dividend and the value of r in the divisor x – r. Example 7: Use synthetic division to divide.  

Your Turn 3: Divide using synthetic division.

Closure Lesson 5-3: Dividing Polynomials Homework: Practice 5.3 part II (worksheet)

Closure Lesson 5-3: Dividing Polynomials Homework: Practice 5.3 part II (worksheet)

(4y4 – 5y2 + 2y + 4) ÷ (2y – 1) long division Warm- up Lesson 5.3 part II (4y4 – 5y2 + 2y + 4) ÷ (2y – 1) long division (x3 - 4x2 + 6x – 4) ÷ (x – 2) synthetic

Part II

(4y4 – 5y2 + 2y + 4) ÷ (2y – 1) long division Warm- up Lesson 5.3 part II Alg. 2B (4y4 – 5y2 + 2y + 4) ÷ (2y – 1) long division 2. (x3 - 4x2 + 6x – 4) ÷ (x – 2) synthetic 2y – 1 4y4 + 0y3 – 5y2 + 2y + 4

Part II

Factoring Polynomials lesson 5-4 A Learning Targets: I can factor polynomials using GCF. I can factor polynomials with four terms by grpouping. I can factor trinomials. I can factor polynomials with two terms.

Technique It means… Example… GCF Grouping Trinomials PST Difference of squares Sum of Cubes Difference of Cubes

Factoring by GCF I can factor a polynomial by using Greatest Common Factor.

3(a + b) = 3a + 3b = x(y – z) = xy – xz = 6y(2x + 1) = 12xy + 6y = Simplifying Factoring 3(a + b) = 3a + 3b = x(y – z) = xy – xz = 6y(2x + 1) = 12xy + 6y =

Example 1: Factor by using Greatest Common Factor (GCF). Find the GCF of each term. Use the GCF to find the remaining factors.

Your Turn 1: Factor by using Greatest Common Factor (GCF). Find the GCF of each term.   Use the GCF to find the remaining factors.

Example 2: Factor by using Greatest Common Factor (GCF). 4x2y – 10x3y4 Your Turn 2: Factor by using Greatest Common Factor (GCF). 14ab3 – 8b3

2x4y2 – 16xy + 24x3y3 15a3b2 + 10a2b4 – 25ab3 Example 3: Factor by using Greatest Common Factor (GCF). 2x4y2 – 16xy + 24x3y3 Your Turn 3: Factor by using Greatest Common Factor (GCF). 15a3b2 + 10a2b4 – 25ab3

Factoring by grouping I can factor polynomials with FOUR terms by grouping .

Example 4: Factor by grouping.

Example 5: Factor by grouping.

Your Turn 4: Factor the following by grouping or using the area model.

Warm up: Simplify: (x3 – 2x2 + 2x – 6) ÷ (x - 3) use synthetic division (3x4 + x3 – 8x2 + 10x – 3) ÷ (3x - 2) use long division Factor: 7xy3 – 14x2y5 + 28x3y2 using GCF ab – 5a + 3b -15 using grouping

Warm up/Review Simplify: (x3 – 2x2 + 2x – 6) ÷ (x - 3) use synthetic division (3x4 + x3 – 8x2 + 10x – 3) ÷ (3x - 2) use long division Factor: 7xy3 – 14x2y5 + 28x3y2 using GCF ab – 5a + 3b -15 using grouping Review: Simplify 5) (x – 3y)(x + 3y) 6) (y – 2)(y + 5) 7) (5 + 2x) + (-1 – x) 8) (-7 – 3x) – (4 – 3x)

Warm up Day 6-Answers Simplify: (x3 – 2x2 + 2x – 6) ÷ (x - 3) x2 + x + 5 + 9/(x-3) (3x4 + x3 – 8x2 + 10x – 3) ÷ (3x - 2) x3 + x2 - 2x + 2 + 1/(3x-2) Factor: 7xy3 – 14x2y5 + 28x3y2 7xy2(y – 2xy3 + 4x2) ab – 5a + 3b -15 (a+3)(b – 5) Review: Simplify 5) (x – 3y)(x + 3y) x2 – 9y2 6) (y – 2)(y + 5) y2 + 3y - 10 7) (5 + 2x) + (-1 – x) 4 + x 8) (-7 – 3x) – (4 – 3x) -11

Part I

Factoring Trinomials lesson 5-4 A I can factor a trinomial.

Technique It means… Example… GCF Grouping Trinomials PST Difference of squares Sum of Cubes Difference of Cubes

whose __________ is __________ . Find factors (product) of ____________ whose __________ is __________ .

Example 1: Factor the trinomial.   a. b.

Your Turn 1: Factor the trinomial. a. b.

Find factors (product) of a & c whose sum is b . REPLACE bx using those two numbers and factor by grouping.

Example 2: Factor the trinomials.   a.

Example 3: Factor the trinomials.   b.

Your Turn 2: Factor the trinomials.   a. b.

If three terms: Factoring Trinomials Example 3: Factor completely. If not possible, write prime. a) b)

Your Turn 2: Factor completely. If not possible, write prime. d) e) f) g) h)

Warm-up/review Lesson 5.1- 5.4: I. Factor completely (lesson 5.4): 1) 40xy + 30x – 100y – 75 2) 3y2 + 21y +36 II. Simplify (lessons 5.1, 5.2 & 5.3): –5(2x)4 4) x5 • x-12 • x 5) (15q6 + 5q2)(5q2)-1 III. Simplify by long division or synthetic: (it’s your choice!) 6) (x3 + 7x2 + 14x + 3) ÷ (x + 2)

Part II

Warm-up/review Lesson 5.1- 5.4: I. Factor completely (lesson 5.4): 1) 40xy + 30x – 100y – 75 2) 3y2 + 21y +36 5(4y + 3)(2x – 5) 3(y + 4)(y + 3) II. Simplify (lessons 5.1, 5.2 & 5.3): –5(2x)4 -80x4 4) x5 • x-12 • x 1 x6 5) (15q6 + 5q2)(5q2)-1 3q4 + 1 III. Simplify by long division or synthetic: (it’s your choice!) 6) (x3 + 7x2 + 14x + 3) ÷ (x + 2) x2 + 5x + 4 + -5 x + 2

Factoring Polynomials lesson 5-4 Objectives: Factor polynomials with three or two terms Identify perfect square trinomials (PST), difference of two squares, & sum or difference of cubes

Special Case of trinomial: PST Go to pg. 20 Example 4: a) b)

I) Difference of two squares Example 5: If two terms: I) Difference of two squares Example 5: b) a) c) d)

If two terms: II) Sum of two cubes Example 6:

If two terms: III) Difference of two cubes Example 7:

Your Turn 3: Factor completely

Your Turn 3: Factor completely Difference of two squares; 4(f + 4)(f – 4) n3 – 125 c. Difference of two cubes; (n – 5)(n2 + 5n + 25)

Simplify quotients of polynomials by factoring Assume no denominator is zero. Example Your Turn

Lesson 5-4: Factoring polynomials Homework: 5.4 (worksheet)

Simplify. 1. 2. 3. 4. Factor the following 5. 6. 7. Warm- up: Lesson 5.4 Alg. 2B Simplify. 1. 2. 3. 4. Synthetic division Factor the following 5. 6. 7.