Adding and Subtracting Polynomials
Polynomial Terminology
Monomial
A monomial is a constant, a variable, or the product of a constant and one or more variables. For example, 5, x, and 2x3y4 are monomials. The exponents of a monomial must be whole numbers.
The degree of a monomial is the sum of the exponents of its variable(s).
| Monomial | Degree |
| 2x3y4 | 3 + 4 = 7 |
| x | 1 |
| 5 | 0 |
Note: The degree of a nonzero constant is always 0.
Polynomial
A polynomial is an expression consisting of one monomial or the sum of monomials.
A binomial is a polynomial consisting of two terms. For example, 7x2 − 5 is a binomial.
A trinomial is a polynomial consisting of three terms. For example, x4 + 10x2 + 25x is an example of a trinomial.
When a polynomial is written in standard form, its monomials are written in descending order with respect to their degrees. The coefficient of the first term of a polynomial written in standard form is referred to as the leading coefficient.
For example:
- The polynomial 2x5 + 6x3 − x2 − 4 is written in standard form. The leading coefficient is 2.
- The polynomial −8x4 + 5x7 − x + 1 is not written in standard form.
The degree of a polynomial is the largest degree of its terms.
| Monomial | Degree |
| 3x5y3 − 11x6 + 12 | 8 |
| x2 + 9x + 20 | 2 |
| 5x4 − 1 | 4 |
Adding Polynomials – Algebra Tiles
Algebra tiles can prove to be a very useful tool when learning how to perform operations on mathematical expressions. In this interactivity, you will learn how to use algebra tiles to model and add polynomial expressions. Click the player button to begin.
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Using Algebra Tiles to Subtract Polynomials
Example: Use algebra tiles to model and simplify the following expression:
(x2 − 4x) − (3x2 − 2x − 1)
Remember that subtraction is the same as adding the opposite. So, the expression (x2 − 4x) − (3x2 − 2x − 1) is equivalent to (x2 − 4x) + (−3x2 + 2x + 1).
Now that the expression is represented as a sum, use algebra tiles to model and simplify the expression
(x2 − 4x) + (−3x2 + 2x + 1).
Group like tiles together.
Eliminate zero pairs.
The expression (x2 − 4x) − (3x2 − 2x − 1) simplifies to −2x2 − 2x + 1.
Finding Sums and Differences of Polynomials
When finding the sums and differences of polynomials, it is important to remember how to perform integer operations and how to combine like terms. In this interactivity, you will discover the steps to adding and subtracting polynomials. Click the player button to begin.
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Practical Problem: Perimeter of a Triangle
In the triangle below, the side lengths are not represented by numerical values. Instead, they are given as polynomial expressions. Read through the example and learn how finding the sum of polynomial expressions can help you represent the perimeter of the triangle.
Example: Find the perimeter of the given triangle.
In your earlier math studies, you learned to determine the perimeter of a figure by finding the sum of the length of its sides. Therefore, you can find the perimeter of this triangle by determining the sum of the polynomial expressions. Specifically, the perimeter of the triangle equals 3x + 4 + 5x + 2 + 4x.
Find the sum by combining like terms:
= 3x + 4 + 5x + 2 + 4x
= 12x + 6
Adding and Subtracting Polynomials Review
Now that you have learned about adding and subtracting polynomials, review your knowledge in this interactivity. Click the player button to get started.
Did you answer the content questions incorrectly? Do you need more instruction or extra practice? If so, view the videos Intro to Polynomials and Adding and Subtracting Polynomials from eMediaVASM.