Algebra 1
Section 8.1 Notes: Adding and Subtracting Polynomials
Polynomial: a monomial or the combinations of monomials, each monomial is a term of the polynomial.
Binomial: the sum of two monomials.
Trinomial: the sum of three monomials.
Degree of a monomial: the sum of the exponents of all its variables. A nonzero constant term has a degree of zero.
Degree of a polynomial: the biggest degree of any term in the polynomial. Polynomials are named based on their degree.

x2 = quadratic
x3 = cubic
x4 = quartic
Example 1: State whether each expression is a polynomial. If it is a polynomial, find the degree and determine whether it is a
monomial, binomial, or trinomial.
Expression
Is it a polynomial?
Degree
a.) 4y – 5xz
b.) –6.5
Yes; 4y – 5xz is the sum of 4y and -5xz
Yes
2
Monomial, binomial, or
trinomial?
Binomial
0
(Constant)
Monomial
---
----
3 (cubic)
Polynomial
c.) 7a-3 + 9b
No; not allowed negative exponents
d.) 6x3 + 4x2 + x + 3
Yes; 4 separate monomial terms
Standard form of a polynomial: the terms are in order from biggest to smallest degree.
Leading coefficient: in standard form, the coefficient of the first term.
Example 2: Write each polynomial in standard form. Identify the leading coefficient (L.C.).
a) 9x2 + 3x6 – 4x
b) –34x + 9x4 + 3x7 – 4x2
3x6 + 9x2 – 4x
3x7 + 9x4 – 4x2 – 34x
L.C.: 3
L.C.: 3
Degree: 6
Degree: 7
Trinomial (3 terms)
Polynomial (4 terms)
Example 3: Add the polynomials. COMBINE LIKE TERMS
a) (7y2 + 2y – 3) + (2 – 4y + 5y2)
12y2 – 2y – 1
b) (4x2 – 2x + 7) + (3x – 7x2 – 9)
-3x2 + x – 2
Example 4: Subtract the polynomials. CHANGE TO ADDITION AND C.L.T.
a) (6y2 + 8y4 – 5y) – (9y4 – 7y + 2y2)
b) (6n2 + 11n3 + 2n) – (4n – 3 + 5n2)
(6y2 + 8y4 – 5y) + (-9y4 + 7y + -2y2)
(6n2 + 11n3 + 2n) + (-4n + 3 + -5n2)
-y4 + 4y2 + 2y
11n3 + n2 – 2n + 3
Example 5: The profit a business makes is found by subtracting the cost to produce an item C from the amount earned in sales S. The
cost to produce and the sales amount could be modeled by the following equations, where x is the number of items produced.
C = 100x2 + 500x – 300
S = 150x2 + 450x + 200
a) Find an equation that models the profit.
Profit = S – C
Profit = (150x2 + 450x + 200) – (100x2 + 500x – 300)
(150x2 + 450x + 200) + (-100x2 + -500x + 300)
50x2 – 50x + 500
b) Use the above equation to predict the profit if 30 items are produced and sold.
Plug x = 30 into all x values in the expression (50x2 – 50x + 500)
50(30)2 – 50(30) + 500
50(900) – 1500 + 500
45000 – 1000
$44,000