naming polynomials 8.1 part 1. what is a polynomial? here are some definitions…
TRANSCRIPT
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Naming Polynomials
8.1Part 1
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What is a Polynomial?
Here are some definitions….
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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
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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.
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Polynomials look like this…
• 4x² + 3x – 1• 8• 9xy²• 3x – 2y• x³• 25x² - 4• 5x³ – 4x + 7
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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
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Names by the number of terms:
1 term : monomialHere are some monomials…3x²7xyx8½x
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2 terms : BinomialHere are some binomials…5x + 13x² - 4x + y
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3 terms : TrinomialHere are some trinomials…7x² + 2x – 10
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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 )
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Examples
Name each expression based on its number of terms
1. 5x + 1 2. 7x²3. 5x – 2xy + 3y4. 6x³ - 9x² + x – 10
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1. 5x + 1 Binomial 2. 7x² Monomial3. 5x – 2xy + 3y Trinomial4. 6x³ - 9x² + x – 10 Polynomial
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Finding Degrees
In order to name a polynomial by degree, you need to know what
degree of a polynomial is, right??
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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 )
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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
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Examples
Find the degree of each polynomial
1. 7x2. x² + 3x – 13. 104. 9x²y³5. 12 – 13x³ + 4x + 5x²
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1. 7x 12. x² + 3x – 1 23. 10 04. 9x²y³ 55. 12 – 13x³ + 4x + 5x² 3
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Names of Polynomials by their Degree
Degree of 0 : ConstantFor example,7
-10
8
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Degree of 1 : LinearFor example,
3x – 2½x + 712x – 1
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Degree of 2 : QuadraticFor example,7x² - 3x + 64x² - 1
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Degree of 3 : Cubic
For example,8x³ + 5x +92x³ - 11
Anything with a degree of 4 or more does not have a special name
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Examples
Name each Polynomial by its degree.
1. 10x³ + 2x2. 3x + 83. 64. 9x² + 3x – 15.
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1. 10x³ + 2x Cubic2. 3x + 8 Linear3. 6 Constant4. 9x² + 3x – 1 Quadratic5. Not a polynomial!
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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
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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
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Standard Form
• STANDARD FORM of a polynomial means that all like terms are combined and the exponents get smaller from left to right.
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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
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1. 4 – 6x³ – 2x + 3x² = -6x³ + 3x² – 2x + 4 cubic/polynomial
2. 3x² - 5x³ + 10 – 7x + x² + 4x = -5x³ + 4x² – 3x + 10 cubic/polynomial
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Summary
Names by Degree• Constant• Linear• Quadratic• Cubic
Names by # of Terms• Monomial• Binomial• Trinomial
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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
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Assignment
Page 373# 1 – 20
Must write problem for credit.No partial credit if incomplete.
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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
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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