Math 101 Intermediate Algebra    Addition, Subtraction, and Multiplication of Polynomials
Chapter 5, Sections 3

Polynomials

General form of a polynomial in x:

anxn + an-1xn-1 + an-2xn-2 + . . . + a2x2 + a1x1 + a0,   where

• the ai, i = 1, 2, . . ., n are real numbers
• n is a whole number

Degree of a term is the sum of the exponents on the variables in the term.

The term 4x3y5 has degree 8 since 3 + 5 = 8.

Degree of a polynomial is the degree of the highest degree term.

To write a polynomial in descending order for a certain variable means to write the polynomial from the term with the highest exponent (in the certain variable) on the left descending to the term with the lowest exponent (in the certain variable) on the right.

Descending order in x:   3x2 - 2x + 1.

Examples of Polynomials in x
Name Example Degree Note
Monomial 3x2 2 One term (mono)
Binomial 2x + 1 1 Two terms (bi)
Trinomial x3 + 2x2 + x 3 Three terms (tri)
Polynomial 6x4 + 5x3 + 4x2 + x + 7 4 Many terms (poly)

Polynomials can be in more than one variable....

Examples of Polynomials in x and y
Name Example Degree Note
Monomial 3x2y3 5 One term (mono)
Binomial 2xy + y2 2 Two terms (bi)
Trinomial x3y4 + 2x3y + xy2 7 Three terms (tri)
Polynomial 6x4y + 5x3y2 + 4x2y3 + xy4 + 7y5 5 Many terms (poly)

Combine like terms.

 Example:   add 3x2 + 2x + 1 and 5x2 - 7x (3x2 + 2x + 1) + (5x2 - 7x)  = =  3x2 + 2x + 1 + 5x2 - 7x remove parentheses =  8x2 - 5x + 1 add like terms

Subtracting Polynomials

1)  Remove parentheses (distribute "-" through).
2)  Combine like terms.

 Example:   subtract 3x2 + 2x + 1 from 5x2 - 7x (5x2 - 7x) - (3x2 + 2x + 1) = =  5x2 + 7x - 3x2 - 2x - 1 remove parentheses =  2x2 + 5x - 1 add like terms

Multiplying Polynomials

1)  Use distributive property to remove parentheses and multiply out.
• FOIL only works when multiplying binomials--the distributive property works when multiplying any polynomials together.
2)  Combine like terms.

 Example:   (2x + 3)(4x + 5) (2x + 3)(4x + 5) = =  2x(4x + 5) + 3(4x + 5) distributive property =  (2x)(4x) + (2x)(5) + 3(4x) + 3(5) distributive property again =  8x2 + 10x + 12x + 15 simplifying =  8x2 + 22x + 15 combine like terms

 Example:   (2x - 3)(4x2 - 5x + 6) (2x - 3)(4x2 - 5x + 6) = =  2x(4x2 - 5x + 6) - 3(4x2 - 5x + 6) distributive property =  (2x)(4x2) - (2x)(5x) + (2x)(6) - 3(4x2) + 3(5x) - 3(6) distributive property again =  8x3 - 10x2 + 12x - 12x2 + 15x - 18 simplifying =  8x3 - 22x2 + 27x - 18 combine like terms

Special Forms

(a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 - 2ab + b2
(a + b)(a - b) = a2 - b2
Note:  the a and b may be any algebraic expression.

Example Note
(x + y)2 = x2 + 2xy + y2 x is a
y is b
(2x + 5)2 =
= (2x)2 + 2(2x)(5) + 52
= 4x2 + 20x + 25
2x is a
5 is b
(2x - 5)2 =
= (2x)2 - 2(2x)(5) + 52
= 4x2 - 20x + 25
2x is a
5 is b
(3x2 - 2y)2 =
= (3x2)2 - 2(3x2)(2y) + (2y)2
= 9x4 - 12x2y + 4y2
3x2 is a
2y is b
(3x + 8)(3x - 8) =
= (3x)2 - 82 =
= 9x2 - 64
3x is a
8 is b
(4x2 - 3y)(4x2 + 3y) =
= (4x2)2 - (3y)2
= 16x4 - 9y2
4x2 is a
3y is b
25y2 - 81x4 =
= (5y)2 - (9x2)2
= (5y + 9x2)(5y - 9x2)
5y is a
9x2 is b