Arithmetic Of Polynomials

Definition of a Polynomial (Vocabulary)

 Definition:   A monomial is a number times a power of x:  axn

Examples

3x2,      1/2 x7,      and      8

are all monomials.

 Definition:   A polynomial  is a sum  or difference of monomials

Examples:

4x5 - 3x2 - 1,    4x2,    2

are all polynomials.

The degree of a polynomial is the largest power of x, the leading coefficient is the number in front of the term with the highest power of x, and the constant term is the number without any x's.

Example:

For the polynomial

4x5 - 3x2 - 1

the degree is 5, the leading coefficient is 4 and the constant term is -1.

Notation:  When we write

P(x) = 3x3 - 2x2 +1

we say "P of x"

To evaluate P(1), we find

3(1)3 - 2(1)2 + 1  =  2

To add or subtract polynomials, we just collect like terms:

Example

Let

P(x)  =  x2 + 3x + 5

and

Q(x)  =  4x3 - 2x2 + 3x - 2

Then

P(x) - Q(x)  =  (x2 + 3x + 5) - (4x3 - 2x2 + 3x - 2)

=  x2 + 3x + 5 - 4x3 + 2x2 - 3x + 2        Distributing the - sign

=  -4x3 + 3x2 + 7                                    Combining like terms

Exercise

Let

P(x)  =  3x2 + 4x - 2

and

Q(x)  =  5x2 - 3x - 5

Find

P(x) + Q(x) Consider the multiplication of the following two first degree polynomials:

(x + 3)(x + 4)

= (x + 3)x + (x + 3)4         Distributing the x + 3

= x2 + 3x + 4x + 12           Distributing the x and the 4

= x2 + 7x + 12                   Combining like terms

Since this type of multiplication occurs so frequently, we have a systematic approach called

FOIL-  Firsts, Outers, Inners, Lasts.

That is we multiply the first terms, the outer terms, the inner terms, and the last terms and add the four results together.

Examples

F      O     I      L

1. (x + 2)(x + 5)  =       x2 +  5x + 2x + 10 = x2 + 7x + 10

2. (3x - 4)(5x + 2)  = 15x2  + 6x - 20x - 8 = 15x2 - 14x - 8

Exercises:

Evaluate the following

1. (x - 2)(3x + 1) 2. (5x + 4)(3x + 2) 3. (3x - y)(2x + 3y) 4. (x + y)(x - y) 5. (x + y)(x + y) 6. (x - y)(x - y) We will note the special products D, E and F as difference of squares, perfect square of sum, and perfect square of difference.

 Special Formulas (a + b)(a - b) = a2 - b2               Difference of Squares (a + b)(a + b) = a2 + 2ab + b2    Square of a Sum (a - b)(a - b) = a2 - 2ab + b2      Square of a Difference

Examples:

1. (3 - x)(3 + x)  =  9 - x2

2. (x + 3)2   =  x2 + 6x + 9

3. (2x - 4)2  =  4x2 - 16x + 16

4. (x + 2)3  =  (x + 2)2(x + 2)  =  (x2 + 4x + 4)(x + 2)

=  (x2 + 4x + 4)x + (x2 + 4x + 4)2

=  x3  + 4x2 + 4x + 2x2  + 8x + 8  =  x3 + 6x2 + 12x + 8.

General Polynomial Multiplication

When the polynomials have more than two terms, we must use the distributive property as follows:

Example

(x3 -3x +1) (x - 3)

=  (x3 -3x +1) (x) + (x3 -3x +1) (-3)

=  x4 - 3x2 + x - 3x3 + 9x - 3

=  x4 - 3x3 -3x2 + 10x - 3