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Learn Polynomials

Add, multiply, factor and divide expressions built from powers of x.

Free to start. Built from a single prompt. No credit card.

A polynomial is a sum of terms made from constants and whole-number powers of a variable, like 4x cubed minus 2x plus 7. Its degree is the highest power, which shapes its graph and limits how many roots it can have. Working with polynomials means adding and multiplying them, factoring them into simpler pieces, dividing one by another, and finding their zeros. These skills sit underneath quadratics, rational functions and calculus. Two results do much of the heavy lifting: the remainder theorem and the factor theorem. Learnr builds a polynomials course at your level and teaches each technique in short interactive lessons with step-by-step checks.

How to learn Polynomials, step by step

Learn the vocabulary precisely. Name the degree, leading coefficient and constant term of examples like 5 - 3x squared + x to the fourth, rewriting each in standard form with powers descending.
Multiply with a grid or distribution. Multiply (x + 2)(x squared - 3x + 1) by distributing each term of the first factor, or with a box grid, then combine like terms. Count terms before combining to avoid losing one.
Practice factoring in a fixed order. Always pull out the greatest common factor first, then look for difference of squares, perfect square trinomials, sum or difference of cubes, and grouping. A fixed checklist prevents missed factors.
Use synthetic division with the factor theorem. For x cubed - 2x squared - 5x + 6, test x = 1: the value is 0, so x - 1 is a factor. Synthetic division leaves x squared - x - 6, which factors as (x - 3)(x + 2).
Connect roots to the graph. Plot the roots, then use the degree and leading coefficient for end behavior. A root of even multiplicity touches the axis and turns, while odd multiplicity crosses it.
Keep factoring sharp with daily sets. Factoring speed decides how hard later algebra feels. Short mixed sets each day in a Learnr course, held steady by a streak, build that speed steadily.

What you'll learn

Identify degree, leading coefficient and standard form
Add, subtract and multiply polynomials
Factor using common factors, grouping and special patterns
Divide polynomials by long and synthetic division
Find roots and sketch graphs from end behavior and multiplicity

What's inside your Polynomials course

A sample path Learnr builds for Polynomials. Your real course adapts to your goal and level.

1. Terms, degree and standard form
2. Adding, subtracting and multiplying
3. Factoring techniques and special products
4. Polynomial long division and synthetic division
5. Remainder theorem, factor theorem and roots
6. Graphs, end behavior and multiplicity

How long does it take to learn Polynomials?

Operations and basic factoring take about 3 to 5 weeks at 3 to 4 hours a week. Division, the factor theorem and graphing higher-degree polynomials add another 3 to 5 weeks, usually within an Algebra 2 or precalculus course.

What you need before you start

Exponent rules, combining like terms, the distributive property and factoring simple quadratics. Comfort with negative numbers matters a great deal here.

Common mistakes to avoid

Forgetting a placeholder zero for a missing power in long or synthetic division, which shifts every column and wrecks the result.
Distributing a minus sign to only the first term when subtracting polynomials. Every term inside the parentheses changes sign.
Writing (x + 3) squared as x squared plus 9. The middle term 6x is required: it equals x squared + 6x + 9.

How Learnr builds your course

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Polynomials questions, answered

Is a square root of x term allowed in a polynomial?

No. Polynomial exponents must be whole numbers 0, 1, 2 and so on, so terms with square roots of x, negative powers or x in a denominator make the expression something other than a polynomial.

How many roots can a polynomial have?

A polynomial of degree n has at most n real roots. Counting complex roots with multiplicity, it has exactly n, which is the fundamental theorem of algebra.