Polynomials


Positive Exponents: If $ a$ is a real number and $ n$ is a natural number, $ a^n$, the nth power of $ a$ is defined as:

$\displaystyle a^n=a\cdot a \cdot \cdot \cdot a$

$ a$ is called the base, $ n$ is called the exponent and $ a^n$ is an exponential expression and is read as '$ a$ to the nth power' or '$ a$ to the nth'.

A rule for exponents For any real number $ a$ and positive integers $ m$ and $ n$,

$\displaystyle a^m\cdot a^n=a^{m+n}$

Ex. $ 2^3\cdot 2^5=2^8$

An Algebraic Expression is the result of adding, subtracting, multiplying, dividing (except by 0), raise to powers, or taking roots on any combination of any variables, such as $ w$, $ x$, $ y$, $ z$, $ a$ and $ b$ or constants, such as -2, 3, 15.

A term is the product of a real number and one or more variables raised to powers. The real number is called the numerical coefficient.

Like terms are terms with the same variables each raised to the same powers.

A polynomial is defined as a term or finite sum of terms with only positive or zero exponents permitted on the variables.

A monomial is a polynomial with exactly one term.

Ex. $ 2x^3w^5$.

A binomial is a polynomial with exactly two terms.

Ex. $ 2x^3w^5+1$.

A trinomial is a polynomial with exactly three terms. Ex. $ 2x^3w^5+1+z$.

A polynomial in one variable $ x$: Ex. $ 1+x+x^2+5x^8$.

Degree of a term in a polynomial in one variable is the exponent on the variable and the degree of a term with more than one variables is the sum of the exponents appearing on the variables.

Degree of a polynomial in one variable is the greatest degree of any term in a polynomial.

Operations on Polynomials:

Addition and Subtraction: Polynomials are added by adding the coefficients of like terms; Polynomials are subtracted by subtracting the coefficients of like terms. Ex. $ (x^2+3x^3)+(4x^3+x^5)=x^2+7x^3+x^5$.

Multiplication: Use FOIL (First, Outside, Inside, Last) to multiply two binomials.

Ex. $ (2x+y)(2x-y)=4x^2-2xy+2xy-y^2=4x^2-y^2$

Division: Ex. $ \frac{x^3+ 2 x^2 -3x + 1}{(x-1)}=(x^2+3x)+\frac{1}{(x-1)}$

\begin{displaymath}\begin{array}{rc@{}c}
 & \multicolumn{2}{l}{      x^2 +3x ...
...mn{2}{l}{      \phantom{1{}-{}}\hspace*{2cm} 1}
 \end{array}\end{displaymath}    



Subsections


Department of Mathematics
Last modified: 2005-08-30