Monomials and polynomials can contain numbers, variables, and exponents. They can be added, subtracted, multiplied, divided, and factored, just like real numbers. There are a few special products of polynomials that are important to know, such as the product of two binomials.
Monomial: A number, a variable with a positive integer exponent, or the product of a number and variable(s) with positive integer exponents.
Polynomial: A monomial or sum of monomials
Term: A part of the polynomial that is added or subtracted.
Coefficient: A number that appears in front of a variable.
Constant: A number without a variable.
Binomial: A polynomial with two terms.
Trinomial: A polynomial with three terms.
Standard Form: A form where the terms in the polynomial are arranged in order of decreasing power (exponents decrease from left to right)
Leading Coefficient: The coefficient of the term with the greatest power.
Degree of a Monomial: Sum of the exponents in the monomial.
Degree of a Polynomial: The greatest degree of the terms.
Like Terms: Terms in the polynomial with the same exponents (coefficients could be different).
Examples of monomials:
These are not monomials:
A polynomial is made up of different terms that contain positive integer powers of the variables.
Example: \((4x^2-3xy+2) + (2x^3+5y) - (x^2+5xy-3)\)
Group like terms: \((2x3) + (4x^2-x^2) + (-3xy-5xy) + (5y) + (2-(-3))\)
Simplify: \(2x^3 + 3x^2 - 8xy + 5y + 5\)
Another method is called FOIL. If given \((a+b)(c+d)\):
So \((a+b)(c+d) = ac + ad + bc + bd\)
Polynomials can be multiplied vertically, similar to vertical multiplication with regular numbers.
Example: \((a+b)(c+d)\)
\(\begin{array}{r}
&a+b\\
\times\!\!\!\!\!\!&c+d\\
\hline
&ad+bd\\
+ac+bc\!\!\!\!\!\!&\\
\hline
&ac+ad+bc+bd\\
\end{array}\)
For example: \(\frac{x^2 + 4x + 5}{x + 3}\)
\(\require{enclose}
\begin{array}{rll}
x+1 && \\[-3pt]
x+3 \enclose{longdiv}{x^2 + 4x +5}\kern-.2ex \\[-3pt]
\underline{-x^2 - 3x\phantom{00}} && \\[-3pt]
x + 5\phantom{0} && \\[-3pt]
\underline{\phantom{0}-x - 3\phantom{0}} && \\[-3pt]
\phantom{0}2 && \\[-3pt]
\end{array}\)
So \(\frac{x^2 + 4x + 5}{x + 3} = x + 1 + \frac{2}{x + 3}\)
Tips:
\((a+b)^2 = (a+b)(a+b)
= a^2+ab+ab+b^2
= a^2+2ab+b^2\)
You can also remember the square of the binomial by drawing this diagram:
The area of the square is \((a+b)(a+b) = (a+b)^2\)
We can also find \((a-b)^2\) by replacing \(b\) with \(-b\):
\((a-b)^2 = (a-b)(a-b)
= a^2-ab-ab-b^2
= a^2-2ab+b^2\)
\((a+b)(a-b) = a^2+ab-ab+b^2
= a^2-b^2\)
\(a\) and \(b\) can represent numbers, variables, or variable expressions.