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Naming Polynomials
8.1Part 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 : monomialHere are some monomials…3x²7xyx8½x
2 terms : BinomialHere are some binomials…5x + 13x² - 4x + y
3 terms : TrinomialHere are some trinomials…7x² + 2x – 10
4 or more terms – polynomialThere 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. 7x²3. 5x – 2xy + 3y4. 6x³ - 9x² + x – 10
1. 5x + 1 Binomial 2. 7x² Monomial3. 5x – 2xy + 3y Trinomial4. 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 DegreeThe degree of a monomial is the sum of the exponents of its variables.
For example,The degree of 7x³ is 3The degree of 8y²z³ is 5The degree of -10xy is 2The 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. 7x2. x² + 3x – 13. 104. 9x²y³5. 12 – 13x³ + 4x + 5x²
1. 7x 12. x² + 3x – 1 23. 10 04. 9x²y³ 55. 12 – 13x³ + 4x + 5x² 3
Names of Polynomials by their Degree
Degree of 0 : ConstantFor example,7
-10
8
Degree of 1 : LinearFor example,
3x – 2½x + 712x – 1
Degree of 2 : QuadraticFor example,7x² - 3x + 64x² - 1
Degree of 3 : Cubic
For example,8x³ + 5x +92x³ - 11
Anything with a degree of 4 or more does not have a special name
Examples
Name each Polynomial by its degree.
1. 10x³ + 2x2. 3x + 83. 64. 9x² + 3x – 15.
1. 10x³ + 2x Cubic2. 3x + 8 Linear3. 6 Constant4. 9x² + 3x – 1 Quadratic5. Not a polynomial!
Putting it all together…ExamplesClassify each polynomial based on its degree and the number of terms:1. 7x³ - 10x 2. 8x – 43. 4x² + 11x – 24. 10x³ + 7x² + 3x – 55. 66. 3x² - 4x
1. 7x³ - 10x cubic/binomial2. 8x – 4 linear/binomial3. 4x² + 11x – 2 quadratic/trinomial4. 10x³ + 7x² + 3x – 5 cubic/polynomial5. 6 constant/monomial6. 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 + 3x²2. 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
Copy the table and fill in the blanks.Polynomial Degree Name by
DegreeNumber of Terms
Name by Terms
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