Section 4.5 – definitions
A Polynomial in one variable is a sum or difference of terms. Each term is of the form axn
where a is a real number and n is a whole number.
Here are a few examples of polynomials:
1) 3x2 + 6x – 5
2)
1
3
𝑦−2
3) 18z5
Here are a few examples of expressions that are NOT polynomials.
1) 4x-2 + 7x – 89
2)
𝑥+3
𝑥−2
Here are a few more definitions.
Monomial – is a polynomial with 1 term.
Examples:
1) 3x
2) 5y2
3) 7 (a monomial doesn’t have to have a letter)
Binomial – is a polynomial with 2 terms.
Examples:
1) 6x + 2
2)
2
5
𝑥 2 − 17𝑥
Trinomial – is a polynomial with 3 terms.
Examples:
1)
11
15
𝑦11 − 12𝑦 + 3𝑦
The coefficient of a term is the number in front of the term.
The degree of a term is the exponent
The following table lists each term together with its coefficient and degree
6x4 + 5x + 7
Term
6x4
5x
7
coefficient
6
5
7
Degree
4
1
0
The Leading Term of a polynomial is the term that has the highest exponent.
For example: the leading term of the polynomial
6x + 3x2 + 5x3 – 2
Is 5x3
The Degree of a polynomial is the exponent of the leading term.
The degree of 6x + 3x2 + 5x3 – 2
is 3, because three is the exponent of the leading term.
The leading coefficient of a polynomial is the coefficient of the leading term.
The leading coefficient of 6x + 3x2 + 5x3 – 2
is 5 because that is the number in front of the leading term.
Polynomials may have several variables. We don’t study them very often.
Here is an example of a polynomial in 3 variables.
7xy – 8z2 + 11 – 5xyz
The individual parts (separated by a plus or minus sign) of a polynomial are called terms.
Like terms are terms with the same variables (letters) raised to the same powers.
These are like terms: 5𝑥 2 ,
1
3
𝑥 2 𝑎𝑛𝑑 𝑥 2
These are not like terms: 5x, 5x2
Like terms can be added and subtracted.
Non like terms can’t be added or subtracted.
These terms can be added, and you add by adding the coefficients:
5x2 + 6x2 + 7x2 =
These terms can’t be added
6x + 7y
These also can’t be added
6x + 7x2