Polynomials Calculator

P(x₀) — Evaluate

Horner's nested method, also P'(x₀) slope

P±Q — Add/Subtract

Align by degree, add/subtract coefficients

P·Q — Multiply

FOIL grid: every term × every term, collect

P÷Q — Long Division

Long division rows + synthetic div + Factor Thm

(x−r) — Factor

GCF → patterns → Δ (quadratic) → RRT + synth

P=0 — Roots

FTA, Descartes' Rule, RRT, bisection, multiplicity

gcd — GCD & LCM

Euclidean algorithm + LCM = P·Q/GCD

P∘Q — Composition

P(Q(x)): build Q^k, sum a_k·Q^k, verify

Horner's \(P(x_0)=(((a_nx_0+a_{n-1})x_0+\cdots)+a_0\)

Division \(P=D\cdot Q+R,\;\deg(R)<\deg(D)\)

Factor Thm \(P(r)=0\Leftrightarrow(x-r)|P(x)\)

Remainder \(P(r)=\text{remainder when }P\div(x-r)\)

FTA \(\deg n\Rightarrow\text{exactly }n\text{ roots}(\mathbb{C})\)

Descartes' \(\text{pos. roots}\leq\text{sign changes in }P\)

Rational Rt \(r=\pm p/q,\;p|a_0,\;q|a_n\)

FOIL \((a+b)(c+d)=ac+ad+bc+bd\)

Diff Sq \(a^2-b^2=(a+b)(a-b)\)

Perf Sq \((a\pm b)^2=a^2\pm2ab+b^2\)

Sum Cubes \(a^3+b^3=(a+b)(a^2-ab+b^2)\)

Diff Cubes \(a^3-b^3=(a-b)(a^2+ab+b^2)\)

GCD·LCM \(\gcd(P,Q)\cdot\text{lcm}=P\cdot Q\)

Euclidean \(\gcd(P,Q)=\gcd(Q,P\bmod Q)\)

Compose deg \(\deg(P\circ Q)=\deg(P)\cdot\deg(Q)\)

Multiplicity \(P(r)=P'(r)=0\Rightarrow\text{mult}\geq2\)

xⁿPolynomials Calculator