Algebra 2 A
Section 5.1 Notes: Operations with Polynomials
To simplify an expression containing powers means to rewrite the expression without parentheses or negative exponents.
Example 1: Simplify each expression.
a)  a 3  a 2 b 4  c 1 
b)
n2
n10
 3a 3 
c)  4 
 b 
2
d)  x 2 y 3  x 3 y 5  z 2 
Polynomials
*A polynomial is an expression that is a sum of variables and exponents.
Ex: 2x3 + x2 – x – 5
*The degree of a polynomial is the degree of the monomial with the greatest degree (the highest exponent)
Ex: The degree of the polynomial above is 3.
*Each piece of the polynomial is called a term.
Ex: 2x3 is a term, x2 is a term, –x is a term, –5 is a term
Names of polynomials
o
o
o
o
Monomial – 1 term
Binomial – 2 terms
Trinomial – 3 terms
Polynomial – 4 or more terms
Degrees of polynomials
o
o
o
o
o
o
Constant – Degree 0
Linear – Degree 1
Quadratic – Degree 2
Cubic – Degree 3
Quartic – Degree 4
Degree n – higher degree
Rules to be a polynomial
 No:
•
•
•
square roots of variables
fractional exponents
variables in the denominator of any fraction
Example 2: Determine whether the given expression is a polynomial. If it is a polynomial, state the degree of the polynomial.
a) c 4  4 c  18
b) 16 p5 
3 2 7
p q
4
c) x2  3x1  7
Adding polynomials: simply combine like terms.
Example 3: Simplify: (4x2 – 9x + 3) + (–2x2 – 5x – 6)
Subtracting polynomials: distribute the negative to the second polynomial and then combine like terms.
Example 4: Simplify: (2a3 + 5a – 7) – (2x3 – 3a + 2)
Example 5: Simplify: (3x2 + 2x – 3) – (4x2 + x – 5)
Multiplying polynomials: distribute OR FOIL/Box method
Example 6: Simplify –y(4y2 + 2y – 3)
Example 7: Simplify (a2 + 3a – 4)(a + 2)
Example 8: Simplify –x(3x2 – 2x + 5)
Example 9: Simplify (x2 + 3x – 2)(x + 4)
Example 10: A small online retailer estimates that the cost, in dollars, associated with selling x units of a particular product is given
by the expression 0.001x2 + 5x + 500. The revenue from selling x units is given by 10𝑥. Write a polynomial to represent the profits
generated by the product if profit = revenue – cost.