Naming Polynomials 8.1 Part 1

Naming Polynomials 8.1 Part 1

What is a Polynomial? Here are some definitions .

Definition of Polynomial An expression that can have constants, variables and exponents, but: * no division by a variable (can’t have something like ) * a variable's exponents can only be 0,1,2,3,. etc (exponents can’t be fractions or negative) * it can't have an infinite number of terms

Here’s another definition A polynomial is a mathematical expression consisting of a sum of terms, each term including a variable or variables raised to a power and multiplied by a coefficient.

Polynomials look like this 4x² 3x – 1 8 9xy² 3x – 2y x³ 25x² - 4 5x³ – 4x 7

Names of Polynomials A Polynomial can be named in two ways It can be named according to the number of terms it has It can be named by its degree

Names by the number of terms: 1 term : monomial Here are some monomials 3x² 7xy x 8 ½x

2 terms : Binomial Here are some binomials 5x 1 3x² - 4 x y

3 terms : Trinomial Here are some trinomials 7x² 2x – 10

4 or more terms – polynomial There is no special name for polynomials with more than 3 terms, so we just refer to them as polynomials (the prefix “poly” means many )

Examples 1. 2. 3. 4. Name each expression based on its number of terms 5x 1 7x² 5x – 2xy 3y 6x³ - 9x² x – 10

1. 2. 3. 4. 5x 1 Binomial 7x² Monomial 5x – 2xy 3y Trinomial 6x³ - 9x² x – 10 Polynomial

Finding Degrees In order to name a polynomial by degree, you need to know what degree of a polynomial is, right?

Finding Degrees Definition of Degree The degree of a monomial is the sum of the exponents of its variables. For example, The degree of 7x³ is 3 The degree of 8y²z³ is 5 The degree of -10xy is 2 The degree of 4 is 0 (since )

The degree of a polynomial in one variable is the same as the greatest exponent. For example, The degree of is 4 The degree of 3x – 4x² 10 is 2

Examples Find the degree of each polynomial 1. 2. 3. 4. 5. 7x x² 3x – 1 10 9x²y³ 12 – 13x³ 4x 5x²

1. 2. 3. 4. 5. 7x 1 x² 3x – 1 2 10 0 9x²y³ 5 12 – 13x³ 4x 5x² 3

Names of Polynomials by their Degree Degree of 0 : Constant For example, 7 -10 8

Degree of 1 : Linear For example, 3x – 2 ½x 7 12x – 1

Degree of 2 : Quadratic For example, 7x² - 3x 6 4x² - 1

Degree of 3 : Cubic For example, 8x³ 5x 9 2x³ - 11 Anything with a degree of 4 or more does not have a special name

Examples Name each Polynomial by its degree. 1. 2. 3. 4. 5. 10x³ 2x 3x 8 6 9x² 3x – 1

1. 2. 3. 4. 5. 10x³ 2x Cubic 3x 8 Linear 6 Constant 9x² 3x – 1 Quadratic Not a polynomial!

Putting it all together Examples Classify each polynomial based on its degree and the number of terms: 1. 7x³ - 10x 2. 8x – 4 3. 4x² 11x – 2 4. 10x³ 7x² 3x – 5 5. 6 6. 3x² - 4x

1. 2. 3. 4. 5. 6. 7x³ - 10x cubic/binomial 8x – 4 linear/binomial 4x² 11x – 2 quadratic/trinomial 10x³ 7x² 3x – 5 cubic/polynomial 6 constant/monomial 3x² - 4x quadratic/binomial

Standard Form STANDARD FORM of a polynomial means that all like terms are combined and the exponents get smaller from left to right.

Examples Put in standard form and then name the polynomial based on its degree and number of terms. 1. 4 – 6x³ – 2x 3x² 2. 3x² - 5x³ 10 – 7x x² 4x

1. 4 – 6x³ – 2x 3x² -6x³ 3x² – 2x 4 cubic/polynomial 2. 3x² - 5x³ 10 – 7x x² 4x -5x³ 4x² – 3x 10 cubic/polynomial

Summary Names by Degree Constant Linear Quadratic Cubic Names by # of Terms Monomial Binomial Trinomial

A word about fractions Coefficients and Constants can be fractions. ½x 5 is ok! -3x² ½ is ok! is not a polynomial is not a polynomial

Assignment Page 373 # 1 – 20 Must write problem for credit. No partial credit if incomplete.

Summary Polynomial Degree Name by Degree Number of Terms Name by Terms Copy the table and fill in the blanks. 7x³ - 2 3 6x² - 10x 1 4x 5

Check yourself! Polynomial Degree Name by Degree Number of Terms Name by Terms 7x³ - 2 3 Cubic 2 Binomial 3 0 Constant 1 Monomial 6x² - 10x 1 2 Quadratic 3 Trinomial 4x 5 1 Linear 2 Binomial