Adding Polynomials Lesson

In this lesson, we will learn how to add and subtract polynomials. Let’s begin by learning about polynomials in general. First and foremost, let’s review the concept of a term. A term is a number, variable, or the product of a number and one or more variables.
3xyz » is a term
-2x²y⁴ » is a term
9 » is a term
In each example above: 3xyz, -2x²y⁴, and 9, we have a single term. When we combine terms together using the addition "+" or subtraction "-" operations we obtain an algebraic expression:
3x³ + 7x² - 5 » algebraic expression
-9x²y - xy + 9 » algebraic expression
When we have a number that multiplies a variable or variables, the number is referred to as a coefficient. When a number is not multiplying a variable, it is known as a constant.
2x + 7 » 2 - coefficient, 7 - constant
-3y + 5 » -3 - coefficient, 5 - constant
When we talk about "like terms", these are terms with the exact same variable parts. This means the variable or variables are the same and each variable is raised to the same power:
3x, 7x » like terms - same variable (x), raised the same power (1)
-5x³, 2x³ » like terms - same variable (x), raised to the same power (3)
9x, 3y » not like terms - different variables (x, y)
11x⁴, 11x³ » not like terms - same variable (x), raised to different powers (4, 3)
We can combine like terms by performing operations with the coefficients and leaving the variable part unchanged. We can show this with the distributive property:
5x + 3x
Add the coefficients (5 and 3) and leave the variable part (x) unchanged:
5x + 3x = (5 + 3)x = 8x
9xy - 13xy
Subtract the coefficients (9 - 13) and leave the variable part (xy) unchanged:
9xy - 13xy = (9 - 13)xy = -4xy

What is a Polynomial?

A polynomial in x (or some other variable) is a single term or the sum of a finite amount of terms (axⁿ) where:
a is any real number
n is any whole number
It is very important to understand that n, the exponent cannot be negative. This will come up early in your studies of polynomials.
5x³ - 2x² + 1 » polynomial
Each term matches the format of axⁿ where a is a real number and n is a whole number. When we think about the constant 1, we can write this as:
1x⁰ = 1 • 1 = 1
Recall raising a number or variable to the power of 0 results in 1. We will see this trick used often in our study of algebra.
5x^-3 - 2x² + 1 » not a polynomial
Notice the exponent of (-3) on x, this violates the definition of a polynomial.
We have special names for polynomials with one term, two terms, and three terms. A polynomial with one term only is known as a monomial. A polynomial with two terms only is known as a binomial. Lastly, a polynomial with three terms is known as a trinomial.
2x² - 11 » binomial (two terms)
-7x⁵ + x² - 5 » trinomial (three terms)
9 » monomial (one term)

Degree of a Polynomial

The degree of a term is the sum of the exponents on all variables of the term.
9x³ » degree of 3
-4x²y³ » degree of 5
7x⁴y⁸z¹¹ » degree of 23
The degree of a polynomial is the largest degree of any non-zero term of the polynomial.
12x⁵ - 2x² + 1 » degree of 5
x⁹y¹² + 2x³y⁴ - 7 » degree of 21

Writing Polynomials in Standard Form

When we work with polynomials, it is custom to write them in standard form. This means the term with the largest degree is first or leftmost, followed by the term with the next largest degree and so on. Let's write the following polynomial in standard form:
7x² + 3 + 2x⁹
We want the term with the highest degree (2x⁹) first:
2x⁹
We follow this by the term with the next largest degree (7x²):
2x⁹ + 7x²
We continue to follow this format, our next and final term will be 3:
2x⁹ + 7x² + 3

How to Add Polynomials

When we add two or more polynomials together, we simply combine like terms. Let's look at a few examples.
Example 1: Simplify each.
(1 + 5x) + (1 + 4x)
To add polynomials, we will just combine like terms. To make this easy, let's drop the parentheses and rearrange the addition:
5x + 4x + 1 + 1
5x + 4x = (5 + 4)x = 9x
1 + 1 = 2
(1 + 5x) + (1 + 4x) = 9x + 2
Example 2: Simplify each.
(2x⁵ - 4) + (x⁵ - 1)
To add polynomials, we will just combine like terms. To make this easy, let's drop the parentheses and rearrange the addition:
2x⁵ + 1x⁵ + (-4) + (-1)
2x⁵ + 1x⁵ = (2 + 1)x⁵ = 3x⁵
(-4) + (-1) = (-5)
(2x⁵ - 4) + (x⁵ - 1) = 3x⁵ - 5
Example 3: Simplify each.
(11x + 11x²) + (7x² - 3x) + (x - x²)
To add polynomials, we will just combine like terms. To make this easy, let's drop the parentheses and rearrange the addition:
11x² + 7x² + (-1x²) + 11x + (-3x) + x
11x² + 7x² + (-1x²) = (11 + 7 + (-1))x² = 17x²
11x + (-3x) + x = (11 + (-3) + 1)x = 9x
(11x + 11x²) + (7x² - 3x) + (x - x²) = 17x² + 9x

How to Subtract Polynomials

When we subtract one polynomial from another, we change the subtraction operation to addition and change each term inside of the parentheses into its opposite. Once this is done, we can add our polynomials by combining like terms. Many teachers will perform this operation by changing the subtraction operation to addition and then placing a (-1) outside of the parentheses. The (-1) will be distributed to each term inside of the parentheses and change the sign of each term.
Example 4: Simplify each.
(14 - 7x) - (2 + 12x)
Let's change our subtraction to addition and change each term inside of the parentheses to its opposite:
(14 - 7x) + (-2 - 12x)
Now we can add, let's drop our parentheses and rearrange terms:
-7x + (-12x) + 14 + (-2)
-7x + (-12x) = (-7 + (-12))x = -19x
14 + (-2) = 12
(14 - 7x) + (-2 - 12x) = -19x + 12
Example 5: Simplify each.
(6x² + 9 - 10x⁴) - (4x⁴ + 6) - (-13 - 9x⁴)
Let's change our subtraction to addition and change each term inside of the parentheses to its opposite:
(6x² + 9 - 10x⁴) + (-4x⁴ - 6) + (13 + 9x⁴)
Now we can add, let's drop our parentheses and rearrange terms:
-10x⁴ + (-4x⁴) + 9x⁴ + 6x² + 9 + (-6) + 13
-10x⁴ + (-4x⁴) + 9x⁴ = (-10 + (-4) + 9)x⁴ =  -5x⁴
6x² » no like terms to combine with
9 + (-6) + 13 = 16
-5x⁴ + 6x² + 16

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