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Adding and Subtracting Polynomials By: Anna Smoak

Adding and Subtracting Polynomials By: Anna Smoak. Warm Up. Definition (in your own words). Facts/Characteristics. A number, a variable, or a product of a number and one or more variables with whole number exponents.

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Adding and Subtracting Polynomials By: Anna Smoak

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  1. Adding and Subtracting PolynomialsBy: Anna Smoak

  2. Warm Up Definition (in your own words) Facts/Characteristics 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 Monomials 3x2y xyz -1 x1/2 a-2 y-2 x + y + z -5x4 12 y ½ b2 Examples Non-examples

  3. 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: x2+ 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

  4. 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:x2+ 2x,   3x3+ 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

  5. Adding Polynomials • How can we represent x2+ 2x + 2using Algebra Tiles? • How can we represent 2x2+ 2xusing Algebra Tiles? • Represent the sum of x2+ 2x + 2and2x2+ 2xusing Algebra Tiles X2 X X 1 + + 1 X2 X2 X X +

  6. x2+ 2x + 2 + 2x2+ 2x X2 X X 1 X2 X2 X X + + + + 1 X2 X2 X2 X X X X 1 + + + + 1 X2 X2 X2 X X X X 1 + + 1 3x2+ 4x + 2

  7. Adding Polynomials • How can we represent -x2+ x + 5using Algebra Tiles? • How can we represent x2+ 2x - 2 using Algebra Tiles? • Draw a picture that represents how we can find the sum of -x2+ x + 5andx2+ 2x - 2 using Algebra Tiles? - X2 X 1 1 1 + + 1 1 X2 X X 1 - + 1

  8. -x2+ x + 5 + x2+ 2x-2 - X2 X 1 1 1 X2 X X 1 - + + + + 1 1 1 - X2 X2 X X X 1 1 1 1 - + + + + 1 1 1 X X X 1 1 + 1 3x+ 3

  9. How can we find the sum of x2 + 2x + 3 and 3x + 1? • Can you think of any other ways we can add polynomials? Combine Like Terms Simplify x2 + 5x + 4 (x2 + 2x + 3) + (3x + 1) = x2 + 2x + 3x + 3 + 1 = What property allows us to get ride of our parentheses?

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

  11. Subtracting Polynomials • How can we represent -(x2+ 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 -(x2+ x + 5) = - x2- x - 5 = • How can we represent (-2x2+ x - 3)using Algebra Tiles? - X2 X 1 1 1 - - 1 1 - X2 X2 X 1 1 - + 1

  12. Subtracting Polynomials • If (- x2- x - 5) = And (-2x2+ x - 3)= How can we find the DIFFERENCE of (- x2- x - 5) and (-2x2+ x - 3)? (- x2- x - 5) - (-2x2+ x - 3)= (- x2- x - 5) + (2x2- x + 3)= which we can now represent with Algebra Tiles - X2 X 1 1 1 - - 1 1 - X2 X2 X 1 1 - + 1

  13. -x2- x - 5 + 2x2- x + 3 - X2 - X - 1 1 1 X2 X2 - X 1 1 + + 1 1 1 X2 X2 X2 - X - X - 1 1 1 1 1 - + + 1 1 1 X2 X X 1 - - 1 x2- 2x - 2

  14. (3x2 – 5x + 3) – (2x2 – x – 4) Simplify Work Steps (3x2 – 5x + 3) – (2x2 – x – 4) = 3x2 – 5x + 3 – 2x2 + x + 4 = 3x2 – 2x2– 5x + x + 3 + 4 = x2– 4x + 7 Distribute The Negative Sign Combine Like Terms Simplify

  15. Find the difference of (– 2x3 + 5x2 – 4x + 8) and (– 2x3 + 3x – 4) (– 2x3 + 5x2 – 4x + 8) - (– 2x3 + 3x – 4) = (– 2x3 + 5x2 – 4x + 8) + ( 2x3 - 3x + 4) = 5x2 – 7x + 12

  16. 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) + (1 + 1 + 2 + 2) Simplifying : Perimeter = 8x + 6

  17. 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.

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