Adding and Subtracting Polynomials By: Anna Smoak

Warm Up Monomials Definition (in your own words)Facts/Characteristics ExamplesNon-examples A number, a variable, or a product of a number and one or more variables with whole number exponents No negative exponents (no exponents in the denominator), no addition or subtraction -5x 4 12 y ½ b 2 3x 2 y xyz x 1/2 a-2 y -2 x + y + z

 Monomials  What does the root word mono- mean?  A monomial contain one term.  Binomials  What does the root word bi- mean?  If monomials contain only one term how many terms do you think a binomial will contain?Example: x 2 + 2x NON-Example: x  Trinomial  What does the root word tri- mean?  If a monomial contains one term and a binomial contains two terms how many terms do you think a trinomial will contain? Example: 4x - 6y + 8 NON-Example: x - y ONE TWO THREE

 Polynomials  What does the root word poly- mean?  How many terms do you think a polynomial will contain?  A polynomial is a monomial or the sum of monomials  Examples: x 2 + 2x, 3x 3 + x² + 5x + 6, 4x - 6y + 8  The ending of these words "nomial" is Greek for "part".  Why is xyx a monomial while x + y + z is a trinomial? xyz is a product of 3 variables with real valued exponents x + y + z is the sum of three monomials MANY

Adding Polynomials  How can we represent x 2 + 2x + 2 using Algebra Tiles?  How can we represent 2x 2 + 2x using Algebra Tiles?  Represent the sum of x 2 + 2x + 2 and 2x 2 + 2x using Algebra Tiles X2X2 X + X X2X2 X2X2 X + X

X 1 + x 2 + 2x x 2 + 2x X2X2 X 1 + X2X2 X2X2 XX + + X2X2 X2X2 X2X2 X + XXX X2X2 X2X2 X2X2 X + X x 2 + 4x XX +

Adding Polynomials X 1 + X2X X2X2 X + X 1 1 -

- x 2 + x x 2 + 2x - 2 3x + 3 X 1 + X2X X2X2 X + X X2X2 - X2X2 + X + X + X XXX

 How can we find the sum of x 2 + 2x + 3 and 3x + 1?  Can you think of any other ways we can add polynomials? What property allows us to get ride of our parentheses? Combine Like TermsSimplify (x 2 + 2x + 3) + (3x + 1) =x 2 + 2x + 3x = x 2 + 5x + 4

 What is my first step in simplifying the expression – (x + 1) ? – (x + 1) = – x – 1 What property did we use to simplify the expression?

Subtracting Polynomials  How can we represent - (x 2 + x + 5) using Algebra Tiles?  Is there anything that we have to do before we can use the Algebra Tiles?  We must distribute the negative sign -(x 2 + x + 5) = - x 2 - x - 5 =  How can we represent (-2x 2 + x - 3) using Algebra Tiles? X 1 X2X X2X2 - X2X2 X

Subtracting Polynomials  If (- x 2 - x - 5) = And (-2x 2 + x - 3)= How can we find the DIFFERENCE of (- x 2 - x - 5) and (-2x 2 + x - 3)? (- x 2 - x - 5) - (-2x 2 + x - 3)= (- x 2 - x - 5) + (2x 2 - x + 3)= which we can now represent with Algebra Tiles X 1 X2X X2X2 - X2X2 X

X - x 2 - x x 2 - x + 3 X2X2 + X2X2 X2X2 X - + X2X2 X2X2 X2X2 XX X2X2 XX 1 1 x 2 - 2x

(3x 2 – 5x + 3) – (2x 2 – x – 4) Simplify WorkSteps (3x 2 – 5x + 3) – (2x 2 – x – 4) = 3x 2 – 5x + 3 – 2x 2 + x + 4 = 3x 2 – 2x 2 – 5x + x = x 2 – 4x + 7 Distribute The Negative Sign Combine Like Terms Simplify

Find the difference of (– 2x 3 + 5x 2 – 4x + 8) and (– 2x 3 + 3x – 4) (– 2x 3 + 5x 2 – 4x + 8) - (– 2x 3 + 3x – 4) = (– 2x 3 + 5x 2 – 4x + 8) + ( 2x 3 - 3x + 4) = 5x 2 – 7x + 12

Write an expression for the perimeter of the rectangle. Then simplify the expression. Perimeter = length + length + width + width Perimeter = (3x + 1) + (3x + 1) + (x + 2) + (x + 2) Combining Like Terms: Perimeter = (3x + 3x + x + x) + ( ) Simplifying : Perimeter = 8x + 6

Ticket out of the door  When two binomials are added, will the sum always, sometimes, or never be a binomial? Explain your answer and give examples to support your answer.