Adding and Subtracting Polynomials March 3, 2014

Standards CCSS.Math.Content.HSA-APR.A.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Objectives Students will be able to identify different polynomials as well as find their degree. Students will be able to add and subtract polynomials.

Key Vocabulary Monomial Binomials Trinomials Polynomials Term Like Terms Degree Standard Form Simplest Form

What is a Monomial? Monomial: An expression that is either a numeral, or the product of a numeral and one or more variables. One term. Examples: 12 (numeral), x (product of 1 and x), 3y (product of 3 and y), -4x2yz (product of -4 and x2yz). Constant Monomial or Constant: a numeral only. Examples: 12, 15, 100

What is a Binomial? Binomial: Two Monomials. Two terms. Examples: 2x – 9 (2x is the first term and -9 is the second term), 2ab + 3xy (2ab is the first term and 3xy is the second term).

What is a Trinomial? Trinomials: Trinomials follow the same concept as the other two but has three terms. Example: x2 + 2x + 3 x2 is the first term, 2x is the second term, and 3 is the last term.

What is a Polynomial? Polynomial: The sum of monomials. Poly- means many. Monomials, binomials, and trinomials are all polynomials

What is a Degree? Degree: The sum of the degree, or exponents. Degree of a Monomial: 3x2y4. Add the exponents of the variables: 2 + 4 = 6. Therefore, the degree of the monomial is 6. Degree of a Polynomial: GREATEST degree of individual monomials Example: Find the degree of the polynomial -4x2y4 + 5x5y3 + 3xy Find the degree of each monomial: 2+4=6, 5+3=8, 1+1=2 The greatest degree is 8, therefore the degree of the polynomial is 8.

Naming Polynomials Name the polynomial by degree and number of terms. Constant Monomial Quartic Monomial Cubic Binomial Quadratic Trinomial Sixth degree polynomial with four terms

Try these! Name the polynomial by degree and number of terms. Sixth degree trinomial Quintic Monomial Linear Monomial Cubic polynomial with four terms Quartic Trinomial

How do we ADD Polynomials? Adding Polynomials: Group like terms together and then add. You may use a variety of organizing techniques to add polynomials which we will explore together. Like Terms: Two or more monomials that have the exact same variable or variables. When simplifying polynomials, you must write your answer in standard form. Standard form is when you only have one variable written in decreasing order of degree.

Adding Method 1: Grouping by Association – Underline each like term with corresponding lines and then add. Get in the habit of using parentheses, it will help out a lot later with subtraction. Now add all of the like terms, same lines, together.

Adding Method 1: Grouping by Association – Underline each like term with corresponding lines and then add. Get in the habit of using parentheses, it will help out a lot later with subtraction. Now add all of the like terms, same lines, together.

Add this on your own

Subtracting Subtraction is the same as addition except for one more step at the beginning. Remember, subtraction is the same as adding a negative so we need to change our subtraction to addition. Change the subtraction to addition and all the signs in the parentheses after the subtraction sign and add as we have done before using any method you choose.

Practice changing the signs and adding.

Practice changing the signs and adding.

The yellow and red squares are additive inverses of each other. Algebra Tiles Algebra tiles can be used to model operations involving integers. Let the small yellow square represent +1 and the small red square (the flip-side) represent -1. The yellow and red squares are additive inverses of each other.

This represents x2 This represents –x2 This represents x This represents -x This represents 1 This represents -1

Zero Pairs Called zero pairs because they are additive inverses of each other. When put together, they cancel each other out to model zero.

Addition of Integers Addition can be viewed as “combining.” Combining involves the forming and removing of all zero pairs.

Addition of Integers 4 (+3) + (+1) = (-2) + (-1) = -3

Addition of Integers 2 (+3) + (-1) = (+4) + (-4) =

Subtraction of Integers Subtraction can be interpreted as “take- away.” Subtraction can also be thought of as “adding the opposite.”

Subtraction of Integers 3 (+5) – (+2) = (-4) – (-3) = -1

Subtracting Integers (+3) – (-5)= (-4) – (+1)= 8 -5

Subtracting Integers (+3) – (-3)= 6

Modeling Polynomials 2x2

More Polynomials Let the blue square represent x2 and the large red square (flip-side) be –x2. Let the green rectangle represent x and the red rectangle (flip-side) represent –x. Let yellow square represent 1 and the small red square (flip-side) represent – 1.

More Polynomials Represent each of the given expressions with algebra tiles. Draw a pictorial diagram of the process. Write the symbolic expression.

More Polynomials 2x + 3= 4x – 2=

More Polynomials Use algebra tiles to simplify each of the given expressions. Combine like terms. Look for zero pairs. Draw a diagram to represent the process. Write the symbolic expression that represents each step.

More Polynomials 2x + 4 + x + 2= -3x + 1 + x + 3=

Try these using Algebra Tiles This process can be used with problems containing x2: