3.2Introduction to Polynomials

Identify Monomials

Monomial or Term: An expression that is a constant or a product of a constant and variables that are raised to whole number powers.

Determine whether the expression is a

monomial5x

-4xy3

X6

8

Determine whether the expression is a

monomial

5x

-4xy3

X6

8

Yes

Yes

Yes

Yes

Monomial or Term: An expression that is a constant or a product of a constant and variables that are

raised to whole number powers.

The following are NOT monomials

5/xy

3x2 + 5x – 4

9a + 5

Identify the coefficient and degree of a

monomialTwo important parts of a monomial are its coefficient and its degree

Coefficient: The numerical factor in a monomial

Degree of a Monomial: The sum of the exponents of all the variables in the monomial

Example Problems

-7x2 Coefficient: -7

Degree: 2

-xy3

Coefficient: -1

Degree: 4

17Coefficient: 17

Degree: 0

Questions ???

Try Some

M6

-7x2y3

14ab

-10

Answers

M6

Coefficient: 1

Degree: 6

-7x2y3

Coefficient: -7

Degree: 5

14ab

Coefficient: 14

Degree: 2

-10

Coefficient: -10

Degree: 0

Identify like terms

Like Terms: Monomials that have the same variables raised to the same exponents.

Note: the definition does not say anything about the coefficient… just the variable and exponent…

Examples

5x2 and 9x2

9xyz and 5xy

7x2y and 7xy2

5a and 9A

Examples

YES

NO

NO

NO

5x2 and 9x2

9xyz and 5xy

7x2y and 7xy2

5a and 9A

You Try…-8x3 and 5x3

12mn and 12m-5y3z and y3z9t and 12T

Identify polynomials and their terms

Polynomial: A monomial or an expression that can be written as a sum of monomials

Simplest form: An expression written with the fewest symbols possible

Polynomial in one variable: A polynomial in which every variable term has the same variable

Multivariable Polynomial: A polynomial with more than one variable

Some Examples Please

Polynomials: 2x3 + 5x + 8

5x + 7

9x3 – 4x2 + 8x – 6

Simplest form:9x3 + (– 4x2) + 8x + (– 6)

9x3 – 4x2 + 8x – 6 Simplest Form

Both acceptable but we prefer simplest form

More ? .....okay

Polynomial in one variable:p3 – 5p +2

12x3 – 8x + 9

Multivariable Polynomial:

q3 + 8q3t

9px2 – 4Px2

Monomials are referred to as terms but polynomials are not because the often have

more than one term. In other words polynomials are made of multiple

monomials.Identify the terms in the polynomial and their coefficients:

9x3 – 4x2 + 8x – 6First Term: 9x3Coefficient: 9

Second Term: - 4x2 Coefficient: - 4

Third Term: 8xCoefficient: 8

Fourth Term: - 6 Coefficient: - 6

Monomial: A single-term polynomialBinomial: A polynomial with exactly two termsTrinomial: A polynomial that has exactly three terms

Polynomials that have more than three terms have no special name

Identify whether the polynomial is a monomial, binomial, trinomial, or has

no special name

5x3

- 3x2 + 5x

4m2 + 6m – 9

x3 + 4x2 – 9x + 2

Identify the degree of a polynomial

The greatest degree of any of the terms in the polynomial.

5x3 + 9x6 – 10x2 + 8x – 2Degree = 6

9x6 has the largest exponent in its term, the degree of the exponent is 6, so 6 is the degree of the polynomial.

Write polynomials in descending order of

degreeTo write a polynomial in descending order of degree, write the term with the greatest degree first, then the term with the next greatest degree, and so on.

5x3 + 4x2 – 3x5 – 7 + 9x

- 3x5 + 5x3 + 4x2 + 9x - 7

9m4 +7m9 – 3m2 + 15m – 6m7 + 2

Your turn…

Practice Worksheet