Introduction to Polynomials
Add and Subtract Polynomials
Introduction to Polynomials
Many applications in mathematics have to do with what are called
polynomials. Polynomials are made up of terms. Terms are a product of
a number (coefficient) and/or a variable(s). For example, 5x, 2y2, -4,
and x are all terms. Polynomial expressions are often named based on
the number of terms which are connected by addition or subtraction.
The term polynomial means many terms.
A monomial is a polynomial that has one term, such as 3x2.
A binomial is a polynomial that has two terms, such as a2 – b2.
A trinomial is a polynomial that has three terms, such as x2 + 4x – 8.
A polynomial larger than three terms does not have a special name.
Polynomials have other key characteristics beside the number of terms
which help us to describe the polynomial.
The leading coefficient is the coefficient of the term with the highest
degree in the polynomial.
The constant term is the term with no variable factors and also has
degree 0.
The degree of the polynomial is the highest degree of any of the terms
in the polynomial.
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Example 1: Identify the key characteristics of 5x
The name of the polynomial is a monomial (one term)
The leading coefficient is 5 (coefficient of term with highest degree)
The constant term is 0 (no non-variable term)
The degree of the polynomial is 1 (highest degree)
Example 2: Identify the key characteristics of 7x4 + 3x3 – 6x2 – x – 5
The name of the polynomial is a polynomial (more than 3 terms)
The leading coefficient is 7 (coefficient of term with highest degree)
The constant term is -5 (term with no variable)
The degree of the polynomial is 4 (highest degree)
Example 3: Identify the key characteristics of x2y2 – 3xy + 7
The name of the polynomial is a trinomial (three terms)
The leading coefficient is 1 (coefficient of term with highest degree)
The constant term is 7 (term with no variable)
The degree of the polynomial is 4 (2+2=4 highest degree)
If we know what the variable in a polynomial represents, we can replace
the variable with the number and evaluate the polynomial.
Example 4: Evaluate 2x2 – 4x + 6 when x = -4
2(-4)2 – 4(-4) + 6
2(16) – 4(-4) + 6
32 + 16 + 6
54
Replace the variable with -4
Simplify the exponent
Simplify
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Example 5: Evaluate -y2 + 2y + 6
-(3)2 + 2(3) + 6
-9 + 2(3) + 6
-9 + 6 + 6
3
when y = 3
Replace the variable with 3
Simplify the exponent
Simplify
Add and Subtract Polynomials
Generally when working with polynomials, we do not know the value of
the variable, so we will try and simplify instead. The simplest operation
with polynomials is addition/subtraction.
When we subtract a
polynomial, we distribute the negative (multiply by -1) through the
polynomial being subtracted and then add. When adding polynomials,
we are merely combining like terms. Rewriting the addition vertically
will put the like terms together.
Example 6: (7x2 – 4x + 3) + (-2x2 – 5x – 1)
7x2 – 4x + 3
-2x2 – 5x – 1
5x2 – 9x + 2
Line up like terms and add the columns
Example 7: (5x2 – 2x + 7) + (3x3 – 9x2 – 11)
5x2 – 2x + 7
3x3 – 9x2
- 11
3
2
3x – 4x – 2x – 4
Line up like terms and add the columns
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Example 8: (4x2 – 2x + 8) – (3x2 + 6x – 4)
(4x2 – 2x + 8) + (-3x2 – 6x + 4) Distribute the negative
4x2 – 2x + 8
-3x2 – 6x + 4
x2 – 8x + 12
Line up like terms and add the columns
Example 9: (2x2 – 4x + 3) + (5x2 – 6x) – (x2 – 9x + 8)
(2x2 – 4x + 3) + (5x2 – 6x) + (-x2 + 9x – 8) Distribute the negative
2x2 – 4x + 3
5x2 – 6x
-x2 + 9x – 8
6x2 – x – 5
Line up like terms and add the columns
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)