1
Operate on polynomials
Add, subtract and multiply polynomials of any degree, and recognise the special products on sight rather than expanding them.
Course · Algebra
Polynomials and Factoring
Multiply out, then learn to go back. The factored form is the one that answers questions — where a graph crosses, when a product is zero — and this course is about producing it reliably rather than by inspection.
(a + b)(a − b) = a² − b²
(a ± b)² = a² ± 2ab + b²
f(c) = 0 ⟺ (x − c) is a factor
root p/q ⟹ p | a₀ and q | aₙ
What this course is about
- Subject
- Algebra
- Lessons
- 13
- Format
- Interactive
Overview
What you will be able to do
By the end you can carry out polynomial arithmetic, factor the standard integer-coefficient forms over the rationals, and state exactly what a completed search has and has not ruled out.
2
Choose the right factoring technique
Work down a decision list — common factor, then term count, then pattern or finite pair search — and expand the result to verify it.
3
Divide polynomials
Carry out long division, use synthetic division where it applies, and read the remainder as a function value.
4
Find every rational root
For an integer-coefficient polynomial, generate every candidate from the constant and leading coefficients, test the list, and distinguish no rational root from no root at all.
Lessons
13 lessons
Defining polynomials and operating on them comes first. “Factoring Out the Greatest Common Factor” is the first move in every factorisation; grouping, trinomial methods and special forms are branches chosen by term count and pattern. The course ends by dividing, finding roots and then sketching.
Polynomials, Degree and Standard FormClassify an expression, standardise it, and report its degree and leading coefficient.Not marked
Adding and Subtracting PolynomialsAdd or subtract by aligning like powers and distributing every subtraction sign.Not marked
Multiplying PolynomialsPredict, distribute every term across every term, collect, and verify the product.Not marked
Special ProductsRecognise and expand the square, conjugate, and cube product patterns.Not marked
Factoring Out the Greatest Common FactorFind the integer-coefficient GCF, factor it out, and expand to check every term.Not marked
Factoring by GroupingTest all pairings of four terms, extract a shared bracket, and keep factoring.Not marked
Factoring Simple TrinomialsExhaust the product-sum pairs for x² + bx + c and defend either result.Not marked
The AC MethodFactor ax² + bx + c by an exhaustive ac pair search followed by grouping.Not marked
Factoring Special FormsTest a polynomial against four special forms, apply a match, and retest every factor.Not marked
Polynomial Long DivisionDivide by any nonzero polynomial, stop by degree, and verify f = dq + r.Not marked
Synthetic Division and the Remainder TheoremUse synthetic division for x - c and verify that its last entry equals f(c).Not marked
The Factor Theorem and Rational RootsGenerate every rational-root candidate, test it exactly, divide out hits, and state the scope of an empty search.Not marked
Graphs of Polynomial FunctionsSketch end behaviour and axis crossings from degree, leading sign, roots, and multiplicities.Not marked
Recommended background
Exponents, distribution, function notation.
Practice suggestions
Always look for a common factor first. The lab's decision list starts there because skipping it is what turns a one-line problem into an unfactorable-looking mess.
Expand your factored answer to check it. Factoring is the only topic on this path where checking is faster than the work itself.
After each complete example, cover the answer and do the faded rehearsal on paper. The first decision is supplied; the remaining algebra and the expansion or substitution check are yours before you answer the quiz.
What it does not cover
Factoring quadratics with irrational or complex coefficients. Quadratics and Complex Numbers teaches the discriminant, the quadratic formula and i; this course stops after proving that no rational factor exists.
A complete factorisation algorithm for arbitrary polynomials of degree 4 or more. No rational root rules out rational linear factors, but a quartic may still split into two quadratics.
Numerical root-finding and polynomial interpolation. An irrational root may be named exactly when its form is already known, but it is not approximated here.