If you’ve ever worked with numbers and variables combined with addition, subtraction, and multiplication, chances are you’ve already encountered polynomials.
But what exactly are they? And why are they so useful in math?
Polynomials are a big part of algebra and help us solve equations, graph lines and curves, and model real-world situations. From simple linear equations to complex higher-degree equations, polynomials allow us to find unknown values.
Beyond algebra, polynomials are also widely used in physics and engineering, guiding scientists in designing everything from rockets to bridges.
In this guide, we’ll explain what polynomials are, explore how to work with them, and practice solving polynomial problems together.
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A polynomial is an algebraic expression that consists of variable and constant terms.
The word “polynomial” comes from the Greek roots “poly-” meaning "many" and the Latin “nomial” meaning "term" or "name." So, a polynomial is an expression with "many terms."
The terms in a polynomial are always separated by addition or subtraction signs, and the exponents of the variables must be non-negative integers (such as 0, 1, 2, 3, and so on).
This means that polynomials cannot have variables with negative exponents, fractional exponents, or variables in the denominator.
An example of a polynomial would be:
4x2 + 7y - 1
Every polynomial consists of one or more terms, and each term has different components.
A term can be a number, a variable, or a combination of both, and they are always separated by addition or subtraction.
In our example, the terms of the polynomial are:
4x2, a squared term
7y, a linear term
-1, a constant
These terms are made up of components which can include:
Coefficient – The numerical factor of the term.
Variable – The letter that represents an unknown value.
Exponent – The power to which the variable is raised.
The key difference between terms and components is that terms are the separate building blocks of an expression, while components are the individual parts within each term.
Think of a term like a slice of pizza, and the components as the cheese, sauce, and toppings that make it complete!
For example, in the term 4x2 (our slice of pizza), the components (or toppings) are:
4 is the coefficient
x is the variable
2 is the exponent
Polynomials come in different forms, and we can classify them based on the number of terms they have or the highest exponent (degree).
Classification by Number of Terms
One way to categorize polynomials is by the number of terms they contain.
1. A monomial is a polynomial with only one term.
This is the simplest type of polynomial with no addition or subtraction. A monomial is just a number, a variable, or both multiplied together.
Examples of monomials:
5x
7x3
-12x2
2. A binomial is a two-term polynomial.
The two terms are separated by either addition or subtraction.
Think of it like two monomials joined together like so:
x2 + 4
3 - 9y
2x - 1
3. Trinomial is a three-term polynomial
A trinomial has three terms, which means it's one step more complex than a binomial.
These are especially common in quadratic equations that look like this:
2a3 + 4a - 7
x2 - 5x + 6
y3 - 3y2 + 8
4. Polynomial has four or more terms
A polynomial is any expression with four or more terms. It’s like a bigger collection of monomials, binomials, or trinomials all put together like this:
x3 + 4x2 - x + 7
5y5 + y2 + 3y - 1
Another way to classify polynomials is by their degree, which is determined by the highest exponent of the variable in the expression.
The degree tells us how complex the polynomial is and helps us understand its behavior when graphed.
A polynomial with no variable, meaning it’s just a number. The graph of a constant polynomial is a horizontal line because its value never changes.
A constant polynomial looks like this: P(x) = c, where c represents the constant.
For instance, in the example P(x) = 5, no matter what value we plug in for x, the polynomial will always equal 5.
A polynomial where the highest exponent is 1. This means the equation represents a straight line when graphed.
Linear polynomials look like this:
6y - 1
3x + 2
A quadratic polynomial is a polynomial where the highest exponent is 2.
When graphed, this type of polynomial forms a parabola. The parabola can open either upward or downward depending on the leading coefficient.
Examples of quadratic polynomials are:
x2 - 5x + 6
y2 + 2y - 1
A polynomial where the highest exponent is 3. These polynomials create curved graphs with turns, often forming an S-shape.
Cubic polynomials look like this:
y3 + 2y
2x3 - 4x
Polynomials follow specific rules that set them apart from other algebraic expressions. These rules help keep polynomials easy to work with while ensuring they follow predictable patterns in algebra.
Polynomials consist of terms that are combined using addition or subtraction.
Now, you may be wondering:
If they are combined using addition or subtraction, how come we have rules for multiplication?
Each term can include constants, variables, and exponents, and they can be multiplied together.
Multiplication of constants and variables is allowed because it does not break the structure of a polynomial because when you multiply constants and variables, you are simply forming new terms.
In polynomial 5x2 + 3x - 7 multiplication is present within terms:
5x2 = 5 × x × x
and
3x = 3 × x
This example follows polynomial rules since each term in the expression is separated by addition or subtraction. The term 5x2 is separated by addition from the 3x term, and 3x term is separated by subtraction from -7 term.
However, division by a variable is not allowed because it can create expressions that become undefined at certain values.
For example, in the expression \(\Large\frac{5}{x}\), if x = 0, the denominator becomes zero, which makes the expression undefined. So, to be considered a polynomial, an expression must not have any variables in the denominator.
The exponent (power) of each variable in a polynomial must be a whole number (0, 1, 2, 3, etc.). This means that polynomials can not have fractions as exponents or negative exponents.
For example, the expression x3 + 5x2 - 4x + 7 is a valid polynomial because all the exponents – 3, 2, and the one we don’t specify, which is 1 – are whole numbers.
However, x-1 + 3x is not a polynomial because it contains a negative exponent (-1) and polynomials cannot have negative exponents.
And x\(\Large\frac{1}{2}\) + 4 is also not a polynomial because it includes a fractional exponent (\(\Large\frac{1}{2}\)), which represents a square root.
To recap: In polynomials, all exponents must be whole numbers.
Now that we know the rules of polynomials, let's look at some examples and determine whether they qualify as polynomials or not, starting with:
4x3 - 2x
Let's check:
Does this expression use only addition, subtraction, and multiplication? Yes! We see subtraction here, which is allowed.
Do the exponents on the variables stay whole numbers? Yes! The exponents are 3 and 1, and both are whole numbers.
Since this expression follows all the polynomial rules, we can confidently say that this is a polynomial!
Now let’s look at:
x-2 + x - 9
At first glance, this might look like a polynomial—it has addition and a constant, which are both fine. But let's take a closer look:
The first term has, x-2, and that negative exponent breaks the rules! Polynomials can only have whole number exponents (like 0, 1, 2, 3…), so this is not a polynomial.
Even though it looks similar to the first example, that little negative exponent makes a big difference!
The next example is:
\(\sqrt{x+5}\)
This example might seem okay because it has addition, which is allowed, but let’s think about what \(\sqrt{x}\) really means:
The square root of 𝑥 is actually the same as writing x\(\Large\frac{1}{2}\). Since the exponent is a fraction 12, instead of a whole number, this means our expression does not qualify as a polynomial.
Finally, let’s check:
x4 - 7x2 + 3x
Let’s go through our checklist again:
Addition and subtraction? Yes!
Only whole number exponents? Yes! The exponents here are 4, 2, and 1, and all of them are whole numbers.
This means our expression follows all the polynomial rules, so this is a valid polynomial!
What have we learned?
If an expression has whole-number exponents, no division by variables, and only addition, subtraction, or multiplication, it’s a polynomial. If it breaks any of these rules, it isn’t.
Ok. We know what polynomials are, we know how to recognize them, now let’s see how to add, subtract, and multiply them.
To add polynomials, we simply combine like terms. Like terms are the terms that have the same variable and exponent. Let’s have a look:
(2x2 - 3x + 1) + (5x2 - x + 4)
Step 1: Find the like terms
Look for the terms that share the same variable and exponent in both polynomials:
x2 terms: 2x2 and 5x2
x terms: -3x and -x
Constant terms: 1 and 4
Step 2: Combine the like terms
Combine the x2 terms. Since both terms have x2, add their coefficients:
2x2 + 5x2 = 7x2
Combine the x terms:
-3x - x = -4x
Combine the constant terms. Add the constants 1 and 4.
1 + 4 = 5
Step 3: Write the final answer
Now, put all the combined terms together:
7x2 - 4x + 5
So the result is:
(2x2 - 3x + 1) + (5x2 - x + 4) = 7x2 - 4x + 5
This process works for all polynomial addition problems—just find like terms and add their coefficients!
To subtract polynomials, we distribute the negative sign to all the terms of the second polynomial and then combine like terms.
Let’s solve:
(4x2 - 2x + 3) - (x2 + 5x - 1)
Step 1: Distribute the Negative Sign
Since we are subtracting the second polynomial, we start by distributing the negative sign across all of its terms. Think of this as multiplying each term inside the parentheses by −1:
4x2 - 2x + 3 - x - 5x + 1
Notice how we lost the brackets? Other changes we made include:
x2 became -x2
5x became -5x
-1 became 1
After this step, the expression now looks like:
Step 2: Find Like terms
Now, let’s group like terms together just like we did in the addition example:
x2 terms: 4x2 and -x2
x terms: -2x and -5x
Constant terms: 3 and 1
Step 3: Subtract the Like Terms
Subtract the x2 terms. Since both terms have x2, subtract their coefficients:
4x2 - x2 = 3x2
Subtract the x terms. The coefficients of the x terms are -2 and -5, so:
-2x - 5x = -7x
Subtract the constants 3 and 1:
3 + 1 = 4
Step 4: Write the Final Answer
Now, put all the combined terms together:
3x2 - 7x + 4
And then write the solved polynomial expression:
(4x2 - 2x + 3) - (x2 + 5x - 1) = 3x2 - 7x + 4
When multiplying polynomials, we use the distributive property, which means we multiply each term in the first polynomial by each term in the second polynomial.
Let’s look at the example:
3x(2x2 + 4x - 5)
Here, we are multiplying a monomial, a single-term expression 3x, by a trinomial, a three-term polynomial, 2x2 + 4x - 5.
Step 1: Distribute 3x to Each Term
This simply means multiply 3x by each term inside the parentheses like so:
3x × 2x2 = 6x3
3x × 4x = 12x2
3x × 5 = -15x
Step 2: Write the solution
We put the products together and the solution to our polynomial is:
3x(2x2 + 4x - 5) = 6x3 + 12x2 - 15x
Polynomials play a role in everything from throwing a ball to designing video games. They help us solve problems in math, science, and everyday life. Scientists, engineers, and computer programmers use them to predict, design, and calculate things in the real world.
One great example of polynomials in action is throwing a ball. When you toss a ball into the air, its height changes over time, and you can use a quadratic polynomial to figure out how high it will go.
A formula like:
h(t) = -5t2 + 20t + 15
shows the ball’s height h(t) at different times t. If you plug in 𝑡 = 2, you can calculate how high the ball is after 2 seconds.
Polynomials also help us find areas and volumes. If you want to find the area of a square, you use the formula x2, where x is the length of a side. If you need to calculate the volume of a cube, you use x3. These simple polynomial equations help in construction, architecture, and design.
Companies use polynomials to predict profits, track population growth, and calculate interest rates. This helps them make better decisions about money and resources.
Saving the best for last:
Have you ever noticed how characters move smoothly in games?
That’s because polynomials help create curves and animations that look realistic. Without polynomials, games and movies wouldn’t look as smooth!
Here are some of the most common questions we hear about polynomials at Mathnasium of Cherry Creek.
The leading term is the term with the highest degree (the largest exponent). The coefficient of this term is called the leading coefficient.
Yes! Polynomials can have positive or negative coefficients. The sign of a coefficient doesn’t determine whether an expression is a polynomial.
Yes! A polynomial can have decimals or fractions in the coefficients (the numbers in front of variables). However, the exponents of variables must be whole numbers.
Example of a polynomial:
2.5x3 - \(\Large\frac{1}{2}\)x + 4
Yes! These are called multivariable polynomials or polynomials in multiple variables.
An example of a multivariable polynomial would be:
3x2y + 2xy3 - y2
Here, the polynomial has two variables: 𝑥 and 𝑦. Each term follows the rules of polynomials.
Yes! When you graph a polynomial, you get a curve that depends on the polynomial’s degree:
Linear polynomials (ax + b) form a straight line.
Quadratic polynomials (ax2 + bx + c) form a parabola.
Cubic polynomials (ax3 + bx2 + cx + d) form an S-shaped curve.
The degree and leading coefficient help determine the shape and direction of the graph.
No, a polynomial must have a finite number of terms.
Even though polynomials can have many terms, they must be countable. An expression with an infinite number of terms is not a polynomial—it would be called a series instead.
A polynomial is a specific type of algebraic expression that follows certain rules:
It only contains whole-number exponents.
It doesn’t have variables in the denominator.
It only uses addition, subtraction, and multiplication (no division by variables or square roots).
A polynomial is an expression made of terms with whole number exponents and no division by variables. A rational expression is a fraction where the numerator and denominator are both polynomials.
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