Polynomials
After studying this lesson, you will be able to:
- Find the degree of a polynomial.
- Classify polynomials.
- Put polynomials in descending order.
A Polynomial is a monomial or a sum of
monomials.
There are 3 Special Names of Polynomials :
Monomials: have one term
Binomials: have two terms
Trinomials: have three terms
**Remember that terms are separated by + and - signs**
Example 1
Classify the polynomial as a monomial, a binomial, or a
trinomial: 3xy
Since this polynomial has one term, it is a monomial .
Example 2
Classify the polynomial as a monomial, a binomial, or a
trinomial: -2x + 4
Since this polynomial has two terms, it is a binomial.
Example 3
Classify the polynomial as a monomial, a binomial, or a
trinomial: x 2 + 2x - 4
Since this polynomial has three terms, it is a trinomial .
Example 4
Classify the polynomial as a monomial, a binomial, or a
trinomial: - 2 x y 2 z 3
Since this polynomial has one term, it is a monomial.
Degree of a Term is the sum of the exponents
of the variables.
The Degree of a Polynomial is the highest
degree of its terms.
Example 5
Identify the degree of each term and the degree of the
polynomial: - 2 x y 2 z 3
This polynomial has one term. To find the degree of the term,
we add the exponents of the variables. The variables are x, y,
and z. The exponents of these variables are 1, 2, and 3. We had
these together to get 6. 6 is the degree of the term . Since
there is only one term, the degree of the polynomial will be 6
also.
Example 6
Identify the degree of each term and the degree of the
polynomial: 9x 6 y 5 - 7x 4 y
3 + 3x 3 y 4 + 17x - 4
This polynomial has five terms. To find the degree of each
term, we add the exponents of the variables. Let's take it one
term at a time.
The degree of the first term will be 11 (we add the exponents
of the variables 5+6=11)
The degree of the second term will be 7 (we add the exponents
of the variables 4+3=7)
The degree of the third term will be 7 (we add the exponents
of the variables 3+4=7)
The degree of the fourth term will be 1 (the only exponent in
this term is 1)
The degree of the fifth term will be 0 (this term has no
variables so its degree is 0)
The degree of the polynomial will be 11 since 11 is the
highest degree of the terms.
Example 7
Identify the degree of each term and the degree of the
polynomial: 8xy + 9x 2 y 2 + 2x 3
y 3
This polynomial has three terms. To find the degree of each
term, we add the exponents of the variables. Let's take it one
term at a time.
The degree of the first term will be 2 (we add the exponents
of the variables 1+1=2)
The degree of the second term will be 4 (we add the exponents
of the variables 2+2=4)
The degree of the third term will be 6 (we add the exponents
of the variables 3+3=6)
The degree of the polynomial will be 6 since 6 is the highest
degree of the terms.
To put a polynomial in Descending Order we
arrange the terms in order from the highest exponent down to the
lowest exponent. We are only concerned with the first variable if
the polynomial has more than one variable.
Example 8
Put the polynomial in descending order for x: 3x + 2x 2
- 4
We need to re-arrange the terms from the highest exponent to
the lowest. 2 is the highest exponent so we put 2x 2
first. The next highest exponent is 1 so we put the 3x next. The
-4 will go last: 2x 2 + 3x - 4
Example 9
Put the polynomial in descending order for x: 6x 2 y
- 4x 3 y 4 - 3x y 2
We need to re-arrange the terms from the highest exponent to
the lowest. We have two variables, but we are only concerned
about the x. 3 is the highest exponent of x so we put - 4x 3
y 4 first. The next highest exponent of x is 2 so we
put the 6x 2 y next. The - 3x y 2 xy will
go last: - 4x 3 y 4 + 6x 2 y -
3x y 2
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