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 Name each expression based on its number of terms 1.5x x² 3.5x – 2xy + 3y 4.6x³ - 9x² + x – 10

1.5x + 1 Binomial 2.7x² Monomial 3.5x – 2xy + 3y Trinomial 4.6x³ - 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 1.7x 2.x² + 3x – x²y³ 5.12 – 13x³ + 4x + 5x²

1.7x 1 2.x² + 3x – x²y³ – 13x³ + 4x + 5x² 3

Names of Polynomials by their Degree Degree of 0 : Constant For example,

Degree of 1 : Linear For example, 3x – 2 ½x x – 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. 1.10x³ + 2x 2.3x x² + 3x – 1 5.

1.10x³ + 2x Cubic 2.3x + 8 Linear 3.6 Constant 4.9x² + 3x – 1 Quadratic 5. 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 – x³ + 7x² + 3x – x² - 4x

1.7x³ - 10x cubic/binomial 2.8x – 4linear/binomial 3.4x² + 11x – 2quadratic/trinomial 4.10x³ + 7x² + 3x – 5cubic/polynomial 5.6constant/monomial 6.3x² - 4xquadratic/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

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

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

Check yourself! PolynomialDegree Name by Degree Number of Terms Name by Terms 7x³ - 23Cubic2Binomial 30Constant1Monomial 6x² - 10x + 12Quadratic3Trinomial 4x + 51Linear2Binomial