Download Lecture notes for Section 5.1

Int. Alg. Notes
Section 5.1
Page 1 of 3
Section 5.1: Adding and Subtracting Polynomials
Big Idea: Polynomials are the most important topic in algebra because any equation that can be written using
addition, subtraction, multiplication, division, integer powers, or roots (which are rational powers) can be
solved by converting the equation into a polynomial equation. The first step toward acquiring this awesome
power is to be able to add and subtract polynomials by combining like terms.
Big Skill: You should be able to add and subtract polynomials by combining like terms.
Example of the kind of equation you’ll be able to solve by the end of chapter 7:
x2  1
2
The equation
  x  2  can be transformed into the polynomial equation x5  6 x 4  38 x 2  80 x  47  0 .
x3
Both of these equations have the same solution of x = 1, as is shown in the graph below:
A monomial in one variable is the product of a constant and a variable raised to a nonnegative integer power.

General form of a monomial in one variable: axk.

The constant a is called the coefficient and k is the degree of the monomial.

Examples of monomials in one variable:
2x2
-3y5
m6
4z
-7
A binomial is the sum of two monomials, like 3p2 – 6p
A trinomial is the sum of three monomials, like p2 – 6p + 9
A polynomial is a monomial or the sum of monomials, like 4p3 + 2p2 + 7p – 12

A polynomial is in standard form when it is written with the terms in descending order of degree.

The degree of the polynomial is the highest degree of any of its terms.
A monomial in many variables is the product of a constant and two or more variables raised to nonnegative
integer powers.

General form of a monomial (in two variables): axmyn

The constant a is called the coefficient and m + n is the degree of the monomial.
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 5.1
Page 2 of 3
To add polynomials, combine like terms.
Practice adding polynomials:
1.  2 x 2  3x  1   4 x 2  5 x  3 
2.
 2z
3
 3 z 2  z    2 z 2  6 z  3 
3.
5x
2
y  3xy 12 xy 2    5 x 2 y  12 xy 5 xy 2  
To subtract polynomials, combine like terms.
Practice subtracting polynomials:
4.  2 x 2  3x  1   4 x 2  5 x  3 
5.
 2z
3
 3 z 2  z    2 z 2  6 z  3 
6.
5x
2
y  3xy 12 xy 2    5 x 2 y  12 xy 5 xy 2  
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 5.1
Page 3 of 3
A polynomial function is a function whose rule is a polynomial.
 The domain of a polynomial function is all real numbers.
 The degree of a polynomial function is the value of the largest exponent on the variable.
 A polynomial function of degree one or zero is called a linear function, like f  x   0.5x 1 , or
f  x  3 .

A polynomial function of degree two is called a quadratic function, like f  x   x2  2 x  9 .

A polynomial function of degree three is called a cubic function, like f  x   2x3  7 x2  5x  3 .
If f and g are two functions, then

The new function that can be made by adding them together is called f + g: (f + g)(x) = f (x) + g(x).
The new function that can be made by subtracting them is called f – g: (f – g)(x) = f (x) – g(x).
Practice:
7. If f  x   2x  9 and g  x   3x2  4 x  7 ; compute (f + g)(x) and (f + g)(2).
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.