Multiplication of Polynomials Using what we already know!

REVIEW What property would be used to simplify the following expression: The Distributive Property

Connecting Recall our definition of a monomial: – A number, variable, or the product of a number and one or more variables How did this differ from a polynomial? – A polynomial adds (or subtracts) 2 or more monomials. By using the distributive property, we have actually multiplied a monomial by a polynomial.

An Alternative Suppose I need to simplify: While distributive property will work, another option is to use generic rectangles! 2x 3x5 Once we find the area of the smaller rectangles, we can add them to get the area of the entire large rectangle. 6x 2 10x The factors are the base and height of the rectangle. Remember Area = base height

PRACTICE Simplify each of the following: Once you have the solutions for each expression, click your mouse again to see the solutions.

Taking the next step What happens if we want to multiply two binomials? – What makes a binomial different from a monomial? A binomial adds 2 monomials together Ex. 2x + 5 Consider the following: When multiplying binomials, we have 2 different methods to choose from: Generic Rectangles FOIL

Generic Rectangles Because we have 4 terms, we need to break the rectangle into 4 sections. Each factor represents the base and the height of the rectangle. x + 5 x + 2 Find the area of the smaller rectangles. x2x2 2x 5x10 Add the areas together to get the total area of the rectangle.

FOIL F: x x = x 2 O: x 2 = 2x I: 5 x = 5x L: 5 2 = 10 The letters in FOIL represents the position of the terms in the expression: First: The x terms are 1 st in each factor. Outside: The x and 2 are on the outside of the expression. Inside: The 5 and x are on the inside of the expression. Last: The 5 and 2 are the last terms in each factor. Once you identify the terms, multiply them. Add your solutions together and simplify. FF O OI ILL

Watch out for the signs How do your answers change when the signs change? (x – 5) (x – 2) (x + 5) (x – 2) (x – 5) (x + 2) Try each of the problems using the method of your choice. Click your mouse to get the solutions.

Find the pattern What is the pattern with with the signs? – When both are positive, the answer has 2 addition signs. (x+5)(x+2) = x 2 +7x+10 – When both are negative, the second sign is negative, the third is positive (x - 5)(x - 2) = x 2 - 7x+10 – When the signs are different, the third sign is negative, the second sign depends on the terms. (x - 5)(x+2) = x 2 - 3x – 10 (x+5)(x - 2) = x 2 +3x - 10

Another example Consider the following: – (2x – 3)(2x + 3) – What do you notice about this problem? – These two factors are known as CONJUGATES Same terms separated by different signs What happens when you multiply two conjugates? – The middle term gets eliminated!! – (2x – 3)(2x + 3)

Practice Multiply the following using your method of choice. When you are finished, Click the mouse again to see the solutions: – (2y + 4)(y – 3) – (-3m+6)(2m+1) – (5n – 2)(n – 7) – (w – 5)(2w + 1)