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| ▭\:\longdivision{▭} | \times \twostack{▭}{▭} | + \twostack{▭}{▭} | - \twostack{▭}{▭} | \left( | \right) | \times | \square\frac{\square}{\square} |
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| - \twostack{▭}{▭} | \lt | 7 | 8 | 9 | \div | AC |
| + \twostack{▭}{▭} | \gt | 4 | 5 | 6 | \times | \square\frac{\square}{\square} |
| \times \twostack{▭}{▭} | \left( | 1 | 2 | 3 | - | x |
| ▭\:\longdivision{▭} | \right) | . | 0 | = | + | y |
Some math feels louder than it should. Terms stacked with exponents, variables pulling in different directions — it can feel like trying to read in a storm. That’s what polynomials are like at first: more noise than meaning. But beneath the confusion, they’re just describing patterns, curves, shifts, movement. And when things get too tangled to untie on your own, Symbolab polynomials calculator steps in. Not to skip the work, but to help you see it more clearly.
At its core, a polynomial is just an expression made of terms. Each term includes a variable (like $x$), sometimes raised to a whole number power, and multiplied by a constant. The terms are added or subtracted — no dividing by variables, no square roots, no fancy tricks. But that’s the definition. Sometimes it helps to see how it actually works.
Imagine this:
You’re helping run a small food stand. For every smoothie sold, you make USD $4$ profit. But there’s also a flat cost of USD $20$ for ingredients and supplies every day. You want to figure out how much profit you’ll make after selling $x$ smoothies.
Your total profit = $4x – 20$
That’s a polynomial, linear, with a variable term $(4x)$ and a constant $(–20)$. It’s clean, predictable, and follows the rules.
Now imagine something a bit more complicated. Let’s say someone offers a strange deal: they’ll pay you USD $10/x$ dollars per smoothie, or USD $3√x$ depending on how many you sell.
That expression would look like: $\frac{10}{x} + 3√x$
This one isn’t a polynomial. Why not? Because:
The difference comes down to structure. Polynomials are built to grow in smooth, predictable curves. They don’t jump, loop, or break. If an expression bends too sharply or involves roots, fractions, or negatives in the exponent, it’s no longer a polynomial.
So next time a problem talks about cost, speed, distance, or growth, look at the shape of the expression. If it stacks up with clean terms and whole number powers, you’re looking at a polynomial.
Not all polynomials are the same. Some are simple, others more complex. The two main things that define them are their degree and the number of terms they have.
The degree of a polynomial is the highest exponent on any variable. It tells you how the polynomial behaves. A higher degree means more bends or changes in the graph. A degree of one creates a straight line. A degree of two curves into a U-shape. Higher degrees start to twist and turn.
Examples:
The type of polynomial depends on how many terms it has. Each term is separated by a plus or minus sign.
| Type | Number of Terms | Example |
|---|---|---|
| Monomial | 1 | $7x^3$ |
| Binomial | 2 | $3x - 5$ |
| Trinomial | 3 | $x^2 + 2x + 1$ |
| Polynomial | 4 or more | $2x^3 - x^2 + 4x - 8$ |
All of them are polynomials. The smaller ones just get special names. Like calling a book a short story or a novel depending on its length.
These names are more than labels. They help describe what kind of math you're looking at. A trinomial might show up when you're calculating area. A cubic polynomial might help model the way a car speeds up. Each one has its place, depending on the shape of the problem.
Understanding degree and type gives you a way to read the expression. It helps you know what to expect before you even start solving.
Polynomials are not just for solving in math class. They are used in everyday situations to describe how things change, move, or grow. Here are some places you’ll find them:
All of these follow patterns that polynomials can describe. When something changes in a steady, measurable way, a polynomial is often close by.
Polynomials don’t always need to be solved. Sometimes the goal is to simplify, combine, or rewrite them in a clearer form. Here are the most common ways to work with polynomials by hand.
Combine like terms — terms that have the same variable and exponent.
Example:
$3x² + 5x – 2 + 4x² – x + 7 $
$= (3x² + 4x²) + (5x – x) + (–2 + 7)$
$ = 7x² + 4x + 5$
Line up like terms and then add or subtract the coefficients.
Example:
$(2x² + 3x + 1) + (x² – x + 4)$
$= 3x² + 2x + 5$
Example (subtraction):
$(4x² + 2x – 1) – (x² + 5x – 3)$
$= (4x² – x²) + (2x – 5x) + (–1 + 3)$
$= 3x² – 3x + 2$
Use the distributive property (also called FOIL when multiplying two binomials).
Example:
$(x + 2)(x + 5) $
$= x(x + 5) + 2(x + 5) $
$= x² + 5x + 2x + 10$
$= x² + 7x + 10$
Factoring is rewriting a polynomial as a product of simpler expressions. Start by checking for a common factor in all terms.
Example:
$6x² + 3x = 3x(2x + 1)$
Or factor a trinomial:
$x² + 5x + 6 = (x + 2)(x + 3)$
Factoring helps prepare for solving, but it also helps with simplifying or graphing.
Polynomials should usually be written in order, from the highest to the lowest degree.
Example:
$4 – x + 2x² → 2x² – x + 4$
This makes it easier to compare, graph, or apply formulas later. Polynomials are like building blocks. You might not always need to solve them, but knowing how to rearrange, combine, and break them apart is a big part of understanding how they work.
Working with polynomials gets easier with practice, but there are a few places where things often go wrong. The good news is, once you spot the patterns in the mistakes, they’re easier to avoid.
Only combine terms that have the exact same variable and exponent.
Example:
$3x² + 4x$ cannot be simplified to $7x$ — they are not like terms.
Tip: Match the variable and the power. If either one is different, leave it alone.
When multiplying, make sure every term gets multiplied.
Example:
$2(x + 3)$ is $2x + 6$, not just $2x$.
Tip: Use arrows or color to track which terms you’ve multiplied. It helps.
Negative signs are easy to lose or flip by mistake.
Example:
$(x – 4)(–2x + 1)$
Forgetting the negative in front of 2x changes everything.
Tip: Put parentheses around negative expressions when subtracting or distributing. It keeps signs from getting lost.
Writing polynomials out of order can make it harder to simplify or compare.
Example: Writing $3 – x + x²$ instead of $x² – x + 3$
Tip: Always write in standard form, from highest to lowest degree.
It’s tempting to go straight to the answer, but skipping steps often leads to small mistakes.
Tip: Write everything down, even if it feels slow. It’s faster than fixing a mistake later.
Trying to factor before the expression is simplified can make it more confusing. Or sometimes factoring is skipped entirely when it could have helped.
Tip: Simplify first, then factor. Don’t guess — look for common patterns.
Sometimes the mistake isn’t in the math, it’s in misunderstanding what the question is asking.
Tip: Take a moment to read the full problem before jumping in. Make sure you're answering what’s actually being asked.
When you’re working with a messy polynomial, maybe one with parentheses, exponents, and multiple variables, it helps to have backup. The Symbolab Polynomials Calculator is more than a shortcut. It’s a learning tool that shows every step, helping you understand the process and check your work. Here’s how to use it.
You can enter your polynomial using any of these methods:
Example to try out: $(x^2 + 2x - 1) - (2x^2 - 3x + 6)$
Once your expression is ready, click the red “Go” button. You don’t need to select anything else at this point.
After you click “Go,” Symbolab will show the complete solution with step-by-step explanations.
You can:
Symbolab Polynomials Calculator isn’t just for checking your answer, it’s for learning how to solve it yourself.
Polynomials are everywhere, from schoolwork to real-world problems. Learning how to simplify, factor, and solve them builds skills you’ll use far beyond math class. The Symbolab Polynomials Calculator helps you learn by showing every step, not just the answer. Use it to build confidence, catch mistakes, and truly understand how polynomial math works.
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