Algebra 2 Multiplying Polynomials

1. Simplify each expression. a.(–4a 3 + a 2 –1) – (–3a 3 – a 2 + 2a + 5) b.(3a 2 – a –6) + (–a 2 – a + 3) 2. Find –2b(3b 3 – b 2 + 5b – 6) Warm-Up Problems

Vocabulary Binomial – An expression consisting of two terms or monomials separated by a plus (+) or minus (−) sign. Examples of binomials include ax + b, x 2 − y 2 and 2x + 3y. Even x is a binomial, since it reduces to x + 9, which has two terms. Trinomial – An algebraic expression consisting of three terms connected by plus or minus signs. Polynomial – An algebraic expression of two or more terms connected by plus or minus signs.

Different Methods of Multiplying Binomials “Distributive” Method “FOIL” “Vertical” Method “Grid” Method Algebra Tiles

"Distributive" Method The most universal method. Applies to all polynomial multiplications, not just to binomials. Multiply (x + 3)(x + 2) Step 1: Multiply the first term of the first binomial with the first term of the second binomial, remembering that when multiplying the values to multiply the bases and add the exponents. Step 2: Multiply the first term of the first binomial with the second term of the second binomial. Step 3: Multiply the second term of the first binomial with the first term of the second binomial. Step 4: Multiply the second term of the first binomial with the second term in the second binomial. Step 5: Put all four values into a single equation by combining like terms.

Use the “Distributive” Method to Solve:

The “FOIL” Method Warning: This method can only be used with binomials!

2x − 7 x + 3 ___________ 6x − 21 2x 2 − 7x ___________ 2x 2 − x − ×32 ____ ____ 672 The “Vertical” Method (2x - 7)(x + 3)(d - 1)(5d - 4)

To multiply by the grid method, place one binomial at the top of a 2x2 grid (for binomials) and the second binomial on the side of the grid. Place the terms such that each term with its sign lines up with a row or column of the grid. Multiply the rows and columns of the grid to complete the interior of the grid. Finish by adding together the entries inside the grid. Answer: x 2 + 5x + 6 The “Grid” Method (3x - 1)(2x + 3)

To multiply binomials using algebra tiles, place one expression at the top of the grid and the second expression on the side of the grid. You MUST maintain straight lines when you are filling in the center of the grid. The tiles needed to complete the inner grid will be your answer. Algebra Tiles (x + 3)(x + 2)

Try It: 1.Use the “Grid” Method to solve (3x – 7)(x + 3). 2.Use Algebra Tiles to solve (2x + 4)(x + 3)

(x – 5)² We know that (x – 5)² is (x – 5)(x – 5) Squaring a Binomial

Try It:

Multiplying Polynomials

Try It:

The diagram shows an area rug on a hardwood floor with dimensions x and x + 3. The rug leaves a border around the outside of the room. The wider borders are 2 feet wide, the narrower ones on the sides are 1 foot wide. a) What is an algebraic expression for the area of the floor? (hint: A = l × w) b) What is an algebraic expression for the area of the rug? c) What algebraic expression represents the area of the border? d) If the value of x is 9 feet, find numerical values for parts a), b) and c). Word Problem

Journal Entry Please answer the following question in your “warm-up” books: Which way do you think is the easiest way to multiply binomials: distributive method, FOIL, vertical method, grid method, or algebra tiles? You must explain your reasoning for your decision.

Additional Practice Problems

Exercises Try the following exercises: 1.(5x − 7)(x + 2) 2. (x 2 + 3)(x 2 + 3) 3. (x − y)(3x − 2y) 4. (4 − y)(y + 4) 5. (a + b)(c + d)

a) (x – 7) 2 =b) (3a – 2b) 2 =c) (2x + y) 2 = d) (2x 3 + 4) 2 =e) (3y – 5z) 2 =f) (6a + 5b) 2 =

a) (2x + 3)(4x – 5) =b) (2 – 7x)(9 + 2x) = c) (9m – 2n)(3m – n) =d) (5x – 3y)(2x + 9y) = e) (xy + 1)(3xy – 1) =f) (2x + 3n)(2x – 3n) = g) (4c – 5d)(c + 2d) =h) (3m 2 – 2n 2 )(2m 2 – n 2 ) = i) (6t + 1)(3t – 2) =j) (2m + 3t)(3m – 4t) =