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Toolbox · Polynomials and factoring

Polynomials and factoring

Expand and collect like terms, factor, divide and evaluate a polynomial, with exact fractional coefficients instead of rounded decimals.

Formula

\begin{gathered} a(x-r_1)(x-r_2)\cdots \\ f(x) = (x-r)q(x) + f(r) \end{gathered}

Factoring turns a sum into a product; the remainder theorem says dividing by (x − r) leaves f(r).

ModePolynomialEvaluate at x₀

Use +, -, * and ^ for powers, for example (x+1)(x-2)(x+3) or (2x-3)^2; only one letter variable is allowed.

Presets

Expanded formx^{3} + 2x^{2} - 5x - 6

Degree3

Leading coefficient1

Degree3

Leading coefficient1

Constant term-6

Value-8

Term	Coefficient
x3	1
x2	2
x1	-5
constant	-6

Standard form: x^{3} + 2x^{2} - 5x - 6
Evaluate: f(1) = -8

Steps

3 / 3

1Formula
$((x+1)(x-2)(x+3))$
Start from the given expression.
2Rule
$x^{3} + 2x^{2} - 5x - 6$
Multiply term by term with the distributive law, then collect like terms.
3Conclusion
$\deg = 3,\quad a_{3} = 1$
The highest power left after collecting is the degree.

Concept

Adding polynomials is collecting like terms and multiplying is repeated distribution. After collecting, the standard form is unique.

Factoring undoes expansion. The result is unique up to a constant factor, so we usually keep going until nothing factors further.

Going further

Why trial roots work: If an integer polynomial has a rational root p/q in lowest terms, then p divides the constant term and q divides the leading coefficient. That leaves only finitely many candidates to test.

Common mistakes

Take out the common factor first; it shrinks the candidate list a lot.
After expanding, check the middle coefficient with the outer-inner sum to catch a missing term.

Page keywordsfactoring polynomialsexpanding bracketspolynomial divisionremainder theoremhow to factor a cubiccross multiplication factoringdifference of squaresperfect square trinomiallong division of polynomialsrational root theorem

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