Algebra 3 Section R.4Polynomials

Objectives of this Section• Recognize Monomials

• Recognize Polynomials• Add, Subtract, and Multiply

Polynomials• Know Formulas for Special Products

A monomial in one variable is the product of a constant times a variable raised to a nonnegative integer power. Thus, a monomial is of the form:

ax k

where a is a constant, x is a variable, and k > 0 is an integer.

Monomial Coefficient Degree

329

4xx

32

-9

Examples of Monomials

A polynomial in one variable is an algebraic expression of the form

a x a x a x ann

nn

11

1 0

Determine the coefficients and degree

of 2 3 54 2x x x .

Example:

Coefficients: 2, 0, -3, 1, -5

Degree: 4

Polynomials are added and subtracted by combining like terms.

Example: Addition

2 8 1 3 5 23 2 3x x x x x

2 3 8 5 1 23 3 2x x x x x

5 3 13 2x x x

Example: Subtraction

2 8 1 3 5 23 2 3x x x x x

2 8 1 3 5 23 2 3x x x x x

2 3 8 5 1 23 3 2x x x x x

x x x3 2 13 3

Polynomial multiplication can be done by using the distributive property multiple times.

Example: Multiplication

3 2 4 32x x x

3 3 4 3 3 2 2 4 2 32 2x x x x x x x

3 12 9 2 8 63 2 2x x x x x

3 14 17 63 2x x x

Special Product FormulasDifference of Two Squares

x a x a x a 2 2

Squares of Binomials, or Perfect Squares

x a x ax a

x a x ax a

2 2 2

2 2 2

2

2

Special Product Formulas

Miscellaneous Trinomials

x a x b x a b x ab 2

ax b cx d acx ad bc x bd 2

Cubes of Binomials, or Perfect Cubes

x a x ax a x a

x a x ax a x a

3 3 2 2 3

3 3 2 2 3

3 3

3 3

Special Product Formulas

Difference of Two Cubes

x a x a x ax a3 3 2 2

Sum of Two Cubes

x a x a x ax a3 3 2 2

Polynomials in Two VariablesThe degree of a polynomial in two variables is the highest degree of all the monomials with nonzero coefficients. The degree of each monomial is the sum of the powers of the variables.

DegreePolynomial2 32x y

2 43 4x xy-

3 2 43 2a ab b- +

Page 43: 1-6 all, 9-57 odds, 67-77 odds

Homework: