Download Polynomials - Yonkers Public Schools

A POLYNOMIAL IS:
An expression that can have
constants, variables and
exponents, that can be combined
using addition, subtraction,
multiplication and division.
A polynomial cannot have division by
a variable.
The variable’s exponent can only be
positive.
A polynomial cannot have an infinite
number of terms.
RELATED VOCABULARY
A term is either a single number or
variable, or numbers and variables
multiplied together.
Monomial: An expression with one term.
This term can be:
A variable : a, x, m , n etc.
A constant : 8, 27, -9 etc.
𝒙
𝟑
Yes, this is allowed. We can divide by a
constant.
BINOMIAL
A binomial is an expression having two
terms separated by addition or
subtraction.
•
•
•
•
•
x+3
2𝑚4 𝑛– 5
3xy + 4
2𝑥 3 - 13
𝑥2 + 𝑦2
TRINOMIAL
A trinomial is an expression having three
terms separated by addition or subtraction.
x+ y + z
10 𝑚2 + 2m – 7
3𝑥 3 - 5𝑥 2 + 9x
𝑥 2 + 5x – 6
Any expression having more than three
terms is just called a polynomial.
So did you notice that the terms in a
polynomial must be unique? That is,
they must be different. In a binomial
there are two different terms.
The three terms in a trinomial are all
different.
So what is meant by different terms?
3𝑥 4 + 3𝑥 3 + 3𝑥 2 + 3𝑥 are all different terms.
Notice that the coefficients and variables are
all the same. However the exponents are
all different.
So if the variables are the same but the
exponents are different then the terms are
different.
DEGREE OF A POLYNOMIAL
The degree of a polynomial is the
highest sum of the exponents of its
variables in any term. For a nonzero
constant the degree is 0. Zero has
no degree.
If the degree of the polynomial is one it is
called linear.
Example : 3x
2x – 7
9y + 4
If the degree of the polynomial is 2 it is
called quadratic.
Example :
4xy + 6
- 5𝑚2 + 2m + 15 (Here the highest exponent
of any of its terms is 2).
If the degree of the polynomial is 3 it is
called cubic.
Example : 8𝑥 2 y
4𝑚3 + 2m – 9
3xyz + 2xy – 5z
After three we just use the ordinal number to
name the degree.
Example :
• fourth degree binomial
-3𝑥 2 𝑦 2 + 7xy
• ninth degree polynomial 𝑥 5 𝑦 4
• eighth degree trinomial
2𝑎3 𝑏5 + 25𝑐 6 - 8