This document discusses polynomials and their properties. It defines polynomials as algebraic expressions involving terms of varying degrees, and defines the degree of a polynomial. It then covers operations like addition, multiplication, and factoring of polynomials using various properties and methods like FOIL. Examples are provided to demonstrate multiplying binomials and special products involving sums and differences of terms.

1. Polynomials

2. The Degree of axn
• If a does not equal 0, the degree of axn is n.
The degree of a nonzero constant is 0. The
constant 0 has no defined degree.

3. Definition of a Polynomial in x
• A polynomial in x is an algebraic
expression of the form
• anxn + an-1xn-1 + an-2xn-2 + … + a1n + a0
• where an, an-1, an-2, …, a1 and a0 are real
numbers. an = 0, and n is a non-negative
integer. The polynomial is of degree n, an is
the leading coefficient, and a0 is the
constant term.

4. Text Example
Perform the indicated operations and simplify:
(-9x3 + 7x2 – 5x + 3) + (13x3 + 2x2 – 8x – 6)
Solution
(-9x3 + 7x2 – 5x + 3) + (13x3 + 2x2 – 8x – 6)
= (-9x3 + 13x3) + (7x2 + 2x2) + (-5x – 8x) + (3 – 6) Group like terms.
= 4x3 + 9x2 – (-13x) + (-3) Combine like terms.
= 4x3 + 9x2 + 13x – 3

5. Multiplying Polynomials
The product of two monomials is obtained by using properties of exponents.
For example,
(-8x6)(5x3) = -8·5x6+3 = -40x9
Multiply coefficients and add exponents.
Furthermore, we can use the distributive property to multiply a monomial and
a polynomial that is not a monomial. For example,
3x4(2x3 – 7x + 3) = 3x4 · 2x3 – 3x4 · 7x + 3x4 · 3 = 6x7 – 21x5 + 9x4.
monomial trinomial

6. Multiplying Polynomials when
Neither is a Monomial
• Multiply each term of one polynomial by
each term of the other polynomial. Then
combine like terms.

7. Using the FOIL Method to Multiply Binomials
(ax + b)(cx + d) = ax · cx + ax · d + b · cx + b ·
d Product of
First terms
Product of
Outside terms
Product of
Inside terms
Product of
Last terms
first
last
inner
outer

8. Text Example
Multiply: (3x + 4)(5x – 3).

9. Text Example
Multiply: (3x + 4)(5x – 3).
Solution
(3x + 4)(5x – 3) = 3x·5x + 3x(-3) + 4(5x) + 4(-3)
= 15x2 – 9x + 20x – 12
= 15x2 + 11x – 12Combine like terms.
first
last
inner
outer
F O I L

10. The Product of the Sum and
Difference of Two Terms
(A  B)(A B)  A2  B2
• The product of the sum and the difference
of the same two terms is the square of the
first term minus the square of the second
term.

11. The Square of a Binomial Sum
(A  B)2  A2  2AB B2
• The square of a binomial sum is first term
squared plus 2 times the product of the
terms plus last term squared.

12. The Square of a Binomial
Difference
(A  B)2  A2  2AB B2
• The square of a binomial difference is first
term squared minus 2 times the product of
the terms plus last term squared.

13. Special Products
Let A and B represent real numbers, variables, or algebraic expressions.
Special Product Example
Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2 (2x + 3)(2x – 3) = (2x) 2 – 32
= 4x2 – 9
Squaring a Binomial
(A + B)2 = A2 + 2AB + B2 (y + 5) 2 = y2 + 2·y·5 + 52
= y2 + 10y + 25
(A – B)2 = A2 – 2AB + B2 (3x – 4) 2 = (3x)2 – 2·3x·4 + 42
= 9x2 – 24x + 16
Cubing a Binomial
(A + B)3 = A3 + 3A2B + 3AB2 + B3 (x + 4)3 = x3 + 3·x2·4 + 3·x·42 + 43
= x3 + 12x2 + 48x + 64
(A – B)3 = A3 – 3A2B – 3AB2 + B3 (x – 2)3 = x3 – 3·x2·2 – 3·x·22 + 23
= x3 – 6x2 – 12x + 8

14. Text Example
Multiply: a. (x + 4y)(3x – 5y) b. (5x + 3y) 2
Solution
We will perform the multiplication in part (a) using the FOIL method. We will
multiply in part (b) using the formula for the square of a binomial, (A + B) 2.
a. (x + 4y)(3x – 5y) Multiply these binomials using the FOIL method.
F O I L
= (x)(3x) + (x)(-5y) + (4y)(3x) + (4y)(-5y)
= 3x2 – 5xy + 12xy – 20y2
= 3x2 + 7xy – 20y2 Combine like terms.
• (5 x + 3y) 2 = (5 x) 2 + 2(5 x)(3y) + (3y) 2 (A + B) 2 = A2 + 2AB + B2
= 25x2 + 30xy + 9y2

15. Example
• Multiply: (3x + 4)2.
( 3x + 4 )2
=(3x)2 + (2)(3x) (4) + 42
=9x2 + 24x + 16
Solution:

16. Polynomials