Lesson 2-2

Warm-up Perform the polynomial operation. 1. (x 2 + 5x – 3) + (x 3 – 2x 2 + 7) 2. (5x – 3 + 2x 2 ) + (4 – 5x 2 + x) 3. (x 2 + 5x – 3) – (x 3 – 2x 2 + 7) 4. (5x – 3 + 2x 2 ) – (4 – 5x 2 + x) Challenge: (4x – 2) – 2(3 – 7x 2 – x)

Refresher Simplify each of the following. 1. 3(x + 5) 2. 7(m – 3) 3. x 2 x x 2 x x 2 5x 3 = 3x + 15 = 7m – 21 = x 5 = 4x 5 = 20x 5

Refresher Perform the polynomial operation. 1. (5x – 3)(2x + 7) 2. (x – 3)(4 + x) 3. (11x + 2)(x + 7) 4. (-x – 5)(3x + 4) Challenge: (4x – 2)(7x 2 – x - 3)

Polynomial Multiplication When we are multiplying polynomials, we are applying the distributive property of multiplication over addition. Pay attention that we only distribute when there is addition happening within the parentheses for which we are multiplying. Example: 2(5x + 3) – use distributive property 10x + 6 5(5xyz) – do not use distributive property 25xyz

Multiplication of Monomial and Polynomial To multiply a polynomial by a monomial, we distribute the monomial to all terms in the polynomial. Example: 6x(x 2 + 5x – 3) 6x x 2 – 18x

Let’s Practice Multiply the following polynomials using distribution. 1. 2x(x + 4) 2. 3x 2 (-2x + x 2 – 7) 3. 2xy(x 2 – 5xy + 3y 2 ) Challenge: 3x(x + 4) + 5(x + 4)

Multiplication Do the following multiplication operations x 35_ x 17_ x 41_ x 81

Multiplication When we are multiplying multi-digit numbers together, we can consider the process similar to that of distribution. We must make sure that we multiply each part of each number by every part of the other number. Example: (35)(62) = (30 + 5)(60 + 2) We have the multiply the 30 by both the 60 and the 2, then multiply the 5 by the 60 and the 2. Finally, we add all of our components to get a final answer. 30(60 + 2) + 5(60 + 2) = 30(60) + 30(2) + 5(60) + 5(2) = 2170

Multiplying Binomials When we multiply binomials, we need to multiply each part of the binomial by each of the other binomial. (You may hear the acronym FOIL to describe this process, but this only works for binomials) Example: (2x + 5)(x + 2) First: (2x)(x) = 2x 2 Outer: (2x)(2) = 4x Inner: (5)(x) = 5x Last: (5)(2) = 10 Finally, we add all the terms: 2x 2 + 4x + 5x + 10 = 2x 2 + 9x + 10

Multiplying with a Grid (3x – 5)(2x + 2)

Multiply Polynomials With polynomials, we need to make sure that we multiply each term by each other. As such, FOILing no longer works because we have more than 2 terms per polynomial. If we set up the problem like a multi-digit multiplication problem, we can progress the same way we always have.

Multiplying Polynomials Find the Product

Multiplying Polynomials with a Grid Find the product (9x 2 – x + 6)(5x – 2)

Let’s Practice Multiply the following polynomials. 1. (m + 3)(5m – 4) 2. (2k – 3)(k 2 + 7k – 8) 3. (a 2 + 5a – 4)(2a + 3) Challenge: (x 2 + 3x + 5)(x 2 – 3x + 5)

Illustrating Binomials with Area Models

Area Model

Example of Area Model Find the product of (x + 4) and (x + 8) using an area model.

Multiplying Polynomials using the area of the polynomial shown You know that the area of a rectangle is the product of its length and width. In the model, let 3x + 1 represent the length and let x + 2 represent the width. To find the total area of the model, add the areas of each rectangular part x X + 2 xxx x xxx x2x2 x2x2 x2x xxx x 1

Multiplying Polynomials using a Volume Model Write a polynomial for the volume of the rectangular prism shown. You know that the volume of a rectangular prism is the product of its length, width, and height. In the figure shown, let x represent the length, x + 1 represent the width and x + 2 represent the height. x + 2 x x + 1

The square of a binomial pattern

a. (7x + 2) 2 b. (3x – 2) 2

Let’s Practice

Sum and Difference Patterns

a. (m + 9)(m – 9)b. (4n – 3)(4n + 3)

Let’s Practice

Expand a binomial Expand the following: (x + 2) 3. Step 1: Rewrite the problem without an exponent. (x + 2)(x + 2)(x + 2) Step 2: Multiply the first two binomials to get a trinomial. (x 2 + 4x + 4)(x + 2) (x+2)(x 2 +4x+4) Step 3: Multiply your new trinomial by the other binomial. x 3 + 4x 2 + 4x + 2x 2 + 8x + 8 Step 4: Combine like terms. x 3 + 6x x + 8

Let’s Practice Expand the following binomial: (x – 3) 3

Homework Pg , #18-27, 30, 48