Naming Polynomials 8 1 Part 1 What is
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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… • • 4 x² + 3 x – 1 8 9 xy² 3 x – 2 y x³ 25 x² - 4 5 x³ – 4 x + 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… 3 x² 7 xy x 8 ½x
2 terms : Binomial Here are some binomials… 5 x + 1 3 x² - 4 x+y
3 terms : Trinomial Here are some trinomials… 7 x² + 2 x – 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 5 x + 1 7 x² 5 x – 2 xy + 3 y 6 x³ - 9 x² + x – 10
1. 2. 3. 4. 5 x + 1 Binomial 7 x² Monomial 5 x – 2 xy + 3 y Trinomial 6 x³ - 9 x² + 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 7 x³ is 3 The degree of 8 y²z³ is 5 The degree of -10 xy 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 3 x – 4 x² + 10 is 2
Examples Find the degree of each polynomial 1. 2. 3. 4. 5. 7 x x² + 3 x – 1 10 9 x²y³ 12 – 13 x³ + 4 x + 5 x²
1. 2. 3. 4. 5. 7 x 1 x² + 3 x – 1 2 10 0 9 x²y³ 5 12 – 13 x³ + 4 x + 5 x² 3
Names of Polynomials by their Degree of 0 : Constant For example, 7 -10 8
Degree of 1 : Linear For example, 3 x – 2 ½x + 7 12 x – 1
Degree of 2 : Quadratic For example, 7 x² - 3 x + 6 4 x² - 1
Degree of 3 : Cubic For example, 8 x³ + 5 x +9 2 x³ - 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. 10 x³ + 2 x 3 x + 8 6 9 x² + 3 x – 1
1. 2. 3. 4. 5. 10 x³ + 2 x 3 x + 8 6 9 x² + 3 x – 1 Cubic Linear Constant Quadratic Not a polynomial!
Putting it all together… Examples Classify each polynomial based on its degree and the number of terms: 1. 7 x³ - 10 x 2. 8 x – 4 3. 4 x² + 11 x – 2 4. 10 x³ + 7 x² + 3 x – 5 5. 6 6. 3 x² - 4 x
1. 2. 3. 4. 5. 6. 7 x³ - 10 x 8 x – 4 4 x² + 11 x – 2 10 x³ + 7 x² + 3 x – 5 6 3 x² - 4 x cubic/binomial linear/binomial quadratic/trinomial cubic/polynomial 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 – 6 x³ – 2 x + 3 x² 2. 3 x² - 5 x³ + 10 – 7 x + x² + 4 x
1. 4 – 6 x³ – 2 x + 3 x² = -6 x³ + 3 x² – 2 x + 4 cubic/polynomial 2. 3 x² - 5 x³ + 10 – 7 x + x² + 4 x = -5 x³ + 4 x² – 3 x + 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! -3 x² + ½ is ok! 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. 7 x³ - 2 3 6 x² - 10 x + 1 4 x + 5
Check yourself! Polynomial Degree Name by Degree Number of Terms Name by Terms 7 x³ - 2 3 Cubic 2 Binomial 3 0 Constant 1 Monomial 6 x² - 10 x + 1 2 Quadratic 3 Trinomial 4 x + 5 1 Linear 2 Binomial
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