Notes 7-3: Polynomials
Remember from 7-1:
A monomial is a number, a variable, or a product
of numbers and variables with whole-number
exponents. A monomial may be a constant or a
single variable.
I. Identifying Polynomials
A polynomial is a monomial or a sum or
difference of monomials.
Some polynomials have special names.
A binomial is the sum of two
monomials. A trinomial is the sum of
three monomials.
• Example: State whether the expression is a polynomial. If it is a
polynomial, identify it as a monomial, binomial, or trinomial.
Expression
Polynomial?
Monomial,
Binomial, or
Trinomial?
2x - 3yz
Yes, 2x - 3yz = 2x + (-3yz), the
sum of two monomials
No, 5n-2 has a negative
exponent, so it is not a
monomial
Yes, -8 is a real number
binomial
8n3+5n-2
-8
4a2 + 5a + a + 9
None of these
Monomial
Yes, the expression simplifies
Monomial
to 4a2 + 6a + 9, so it is the sum
of three monomials
II. Degrees and Leading
Coefficients
The terms of a polynomial are the
monomials that are being added or
subtracted.
The degree of a polynomial is the
degree of the term with the greatest
degree.
The leading coefficient is the
coefficient of the variable with the
highest degree.
Find the degree and leading coefficient of
each polynomial
Polynomial
Terms
Degree
5n2
5n2
2
-4x3 + 3x2 + 5
-4x2, 3x2, 3
5
-4
-a4-1
-a4, -1
-1
4
Leading
Coefficient
5
III. Ordering the terms of a polynomial
The terms of a polynomial may be
written in any order. However, the
terms of a polynomial are usually
arranged so that the powers of one
variable are in descending
(decreasing, large to small) order.
Examples: Arrange the terms of each polynomial so that the
powers of x are in descending order.
A. 7x2 + 2x4 - 11
2x4 + 7x2 – 11x
B. 2xy3 + y2 + 5x3 – 3x2y
2x1y3 + y2 + 5x3 – 3x2y1
5x3 -3x2y + 2xy3 +y2
Arrange the terms of each polynomial so that the powers of
x are in descending order.
C. 6x2 + 5 – 8x -2x3
-2x3 + 6x2– 8x + 5
D. 3a3x2 – a4 + 4ax5 + 9a2x
4ax5 + 3a3x2 + 9a2x – a4
You Try!
Arrange the terms of each polynomial so that
the powers of x are in descending order.
E. 3x2y4 + 2x4y2 – 4x3y + x5 – y2
x5+ 2x4y2– 4x3y + 3x2y4 -y2