Understanding Polynomials: Types and Naming Conventions
Delve into the world of polynomials with definitions, examples, and explanations of how they are named based on the number of terms they contain. Learn about monomials, binomials, trinomials, and general polynomials, along with insightful images to aid your understanding.
Download Presentation
Please find below an Image/Link to download the presentation.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author. Download presentation by click this link. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
E N D
Presentation Transcript
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
Heres 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 4x + 3x 1 8 9xy 3x 2y x 25x - 4 5x 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 3x 7xy x 8 x
2 terms : Binomial Here are some binomials 5x + 1 3x - 4 x + y
3 terms : Trinomial Here are some trinomials 7x + 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 + 1 2. 7x 3. 5x 2xy + 3y 4. 6x - 9x + x 10
1. 5x + 1 Binomial 2. 7x Monomial 3. 5x 2xy + 3y Trinomial 4. 6x - 9x + 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 7x is 3 The degree of 8y z 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 4x + 10 is 2
Examples Find the degree of each polynomial 1. 7x 2. x + 3x 1 3. 10 4. 9x y 5. 12 13x + 4x + 5x
1. 7x 1 2. x + 3x 1 2 3. 10 0 4. 9x y 5 5. 12 13x + 4x + 5x 3
Names of Polynomials by their Degree Degree of 0 : Constant For example, 7 -10 8
Degree of 1 : Linear For example, 3x 2 x + 7 12x 1
Degree of 2 : Quadratic For example, 7x - 3x + 6 4x - 1
Degree of 3 : Cubic For example, 8x + 5x +9 2x - 11 Anything with a degree of 4 or more does not have a special name
Examples Name each Polynomial by its degree. 1. 10x + 2x 2. 3x + 8 3. 6 4. 9x + 3x 1 5.
1. 10x + 2x 2. 3x + 8 3. 6 Constant 4. 9x + 3x 1 Quadratic 5. Not a polynomial! Cubic Linear
Putting it all together Examples Classify each polynomial based on its degree and the number of terms: 1. 7x - 10x 2. 8x 4 3. 4x + 11x 2 4. 10x + 7x + 3x 5 5. 6 6. 3x - 4x
1. 7x - 10x 2. 8x 4 3. 4x + 11x 2 4. 10x + 7x + 3x 5 cubic/polynomial 5. 6 6. 3x - 4x cubic/binomial linear/binomial quadratic/trinomial constant/monomial 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 6x 2x + 3x 2. 3x - 5x + 10 7x + x + 4x
1. 4 6x 2x + 3x = -6x + 3x 2x + 4 cubic/polynomial 2. 3x - 5x + 10 7x + x + 4x = -5x + 4x 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. x + 5 is ok! -3x + 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. 7x - 2 3 6x - 10x + 1 4x + 5
Check yourself! Polynomial Degree Name by Degree Number of Terms Name by Terms 7x - 2 3 Cubic 2 Binomial 3 0 Constant 1 Monomial 6x - 10x + 1 2 Quadratic 3 Trinomial 4x + 5 1 Linear 2 Binomial
