A term is an algebraic expression that is either a constant or a product of a constant and one or more variables raised to whole-number powers.

Examples of terms include: 9x, 2xy, 5

A polynomial is a finite sum of one or more terms. Examples of polynomials are:

The degree of a term is the sum of the exponents of its variables. The degree of a nonzero constant is zero. For example, 2x has degree 6; has degree 7; has degree 6 (s=s)

The degree of a polynomial is the highest degree of any of its terms. For example,

has degree 5; has degree 14 (7+5+2=14)

A trinomial is a polynomial containing exactly three terms.

A binomial is a polynomial containing exactly two terms.

A monomial is a polynomial containing exactly one term.

Polynomials are added by adding coefficients of like terms.

Polynomials are subtracted by subtracting coefficients of like terms.

Example.

a) Add:

b) Subtract:

Solution:

a)

b)

To multiply polynomials, multiply each term of one polynomial by each term of the other polynomial and then combine like terms.

Special Products

(x + y)(x – y) = x - y

(x + y) = x + 2xy + y

(x – y) = x - 2xy + y

The FOIL Method

When multiplying two binomials, the letters of FOIL can help you remember which terms are multiplied to complete the product.

F – firsts Multiply the first terms of the binomials

O – outers Multiply the outer terms of the binomials

I – inners Multiply the inner terms of the binomials

L – lasts Multiply the last terms of the binomials

For example, F O I L

(2x – 1)(3x + 5) = (2x)(3x) + (2x)(5) + (-1)(3x) + (-1)(5)

= 6x + 10x – 3x – 5

= 6x+ 7x – 5

Example.

Multiply: a) (2x – 3)(x + 3x – 5)

b) (3x + 5)(3x – 5)

c) (5x – 6)(4x + 3)

d) (x + 3)

e) (7x – 4)

Solution:

a) (2x – 3)(x+ 3x – 5) = 2x(x+ 3x – 5) – 3(x + 3x – 5)

= 2x(x) + 2x(3x) + 2x(-5) – 3(x) – 3(3x) –3(-5)

= 2x + 6x - 10x – 3x - 9x + 15

= 2x + 3x - 19x + 15

b) (3x + 5)(3x – 5) = (3x) - 5 Special Product: (x+y)(x-y) = x - y

= 9x - 25

F O I L

c) (5x – 6)(4x + 3) = (5x)(4x) + (5x)(3) + (-6)(4x) + (-6)(3)

= 20x + 15x – 24x - 18

= 20x - 9x – 18

d) (x + 3) = x + 2(x)(3) + 3 Special Product: (x+y)= x+2xy+y

= x + 6x + 9

e) (7x – 4) = (7x) - 2(7x)(4) + 4 Special Product: (x-y) = x- 2xy + y

= 49x - 56x + 16