 Polynomials
 General form of a polynomial in x:

a_{n}x^{n} + a_{n1}x^{n1} +
a_{n2}x^{n2} + . . . +
a_{2}x^{2} + a_{1}x^{1} + a_{0}, where
 the a_{i}, 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 4x^{3}y^{5} 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: 3x^{2}  2x + 1.

Examples of Polynomials in x 
Name 
Example 
Degree 
Note 
Monomial 
3x^{2} 
2 
One term (mono) 
Binomial 
2x + 1 
1 
Two terms (bi) 
Trinomial 
x^{3} + 2x^{2} + x 
3 
Three terms (tri) 
Polynomial 
6x^{4} + 5x^{3} + 4x^{2} + 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 
3x^{2}y^{3} 
5 
One term (mono) 
Binomial 
2xy + y^{2} 
2 
Two terms (bi) 
Trinomial 
x^{3}y^{4} + 2x^{3}y + xy^{2} 
7 
Three terms (tri) 
Polynomial 
6x^{4}y + 5x^{3}y^{2} + 4x^{2}y^{3} + xy^{4} + 7y^{5} 
5 
Many terms (poly) 
 Adding Polynomials
 Combine like terms.
Example: add 3x^{2} + 2x + 1 and 5x^{2}  7x 
(3x^{2} + 2x + 1) + (5x^{2}  7x) = 

= 3x^{2} + 2x + 1 + 5x^{2}  7x 
remove parentheses 
= 8x^{2}  5x + 1 
add like terms 
 Subtracting Polynomials
 1) Remove parentheses (distribute "" through).
 2) Combine like terms.
Example: subtract 3x^{2} + 2x + 1 from 5x^{2}  7x 
(5x^{2}  7x)  (3x^{2} + 2x + 1) = 

= 5x^{2} + 7x  3x^{2}  2x  1 
remove parentheses 
= 2x^{2} + 5x  1 
add like terms 
 Multiplying Polynomials
 1) Use distributive property to remove parentheses and multiply out.
 FOIL only works when multiplying binomialsthe 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 
= 8x^{2} + 10x + 12x + 15 
simplifying 
= 8x^{2} + 22x + 15 
combine like terms 
Example: (2x  3)(4x^{2}  5x + 6) 
(2x  3)(4x^{2}  5x + 6) = 

= 2x(4x^{2}  5x + 6)  3(4x^{2}  5x + 6) 
distributive property 
= (2x)(4x^{2})  (2x)(5x) + (2x)(6)  3(4x^{2}) + 3(5x)  3(6) 
distributive property again 
= 8x^{3}  10x^{2} + 12x  12x^{2} + 15x  18 
simplifying 
= 8x^{3}  22x^{2} + 27x  18 
combine like terms 
 Special Forms
 (a + b)^{2} = a^{2} + 2ab + b^{2}
 (a  b)^{2} = a^{2}  2ab + b^{2}
 (a + b)(a  b) = a^{2}  b^{2}
 Note: the a and b may be any algebraic expression.
Example 
Note 
(x + y)^{2} = x^{2} + 2xy + y^{2} 
x is a y is b 
(2x + 5)^{2} =
= (2x)^{2} + 2(2x)(5) + 5^{2}
= 4x^{2} + 20x + 25 
2x is a 5 is b 
(2x  5)^{2} =
= (2x)^{2}  2(2x)(5) + 5^{2}
= 4x^{2}  20x + 25 
2x is a 5 is b 
(3x^{2}  2y)^{2} =
= (3x^{2})^{2}  2(3x^{2})(2y) + (2y)^{2}
= 9x^{4}  12x^{2}y + 4y^{2} 
3x^{2} is a 2y is b 
(3x + 8)(3x  8) =
= (3x)^{2}  8^{2} =
= 9x^{2}  64 
3x is a 8 is b 
(4x^{2}  3y)(4x^{2} + 3y) =
= (4x^{2})^{2}  (3y)^{2}
= 16x^{4}  9y^{2} 
4x^{2} is a 3y is b 
25y^{2}  81x^{4} =
= (5y)^{2}  (9x^{2})^{2}
= (5y + 9x^{2})(5y  9x^{2}) 
5y is a 9x^{2} is b 
