Polynomials

Polynomial structure and degree, roots and factors (including remainder and factor theorems), division with linear divisors, and relationships among roots, coefficients and conjugate pairs.

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Polynomial Basics

Polynomial definition in x with coefficients and degree, identifying terms like akxk, finding degrees of polynomial products and specific coefficients in products, and matching coefficients in polynomial identities that hold for all x∈R.

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Polynomial division, factors and remainders

Polynomial division with quotient and remainder, including long division by a linear divisor, the degree condition that degR<degD, and the remainder and factor theorems (P(a) is the remainder on division by (x−a), and (x−a) is a factor iff P(a)=0).

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Polynomial graphs

Polynomial graphs and how the sign of the leading coefficient affects the broad shape of cubic and quartic graphs, and how real roots appear as x-intercepts where the curve touches or crosses the axis depending on whether the root is repeated or unique.

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Polynomial roots

A degree n polynomial has n roots counting multiplicity, and its roots may be real or complex; for a polynomial with real coefficients, complex roots occur in conjugate pairs, and the sum and product of roots are given by the coefficient relationships an−an−1 and an(−1)na0.

0/0 key exercises

Polynomials

Polynomial structure and degree, roots and factors (including remainder and factor theorems), division with linear divisors, and relationships among roots, coefficients and conjugate pairs.

Sign up to view your learning progress

Sign up

Learning path

Follow the recommended order, or open any concept.

Start Here

Polynomial Basics

Polynomial definition in x with coefficients and degree, identifying terms like akxk, finding degrees of polynomial products and specific coefficients in products, and matching coefficients in polynomial identities that hold for all x∈R.

0/0 key exercises

Polynomial division, factors and remainders

Polynomial division with quotient and remainder, including long division by a linear divisor, the degree condition that degR<degD, and the remainder and factor theorems (P(a) is the remainder on division by (x−a), and (x−a) is a factor iff P(a)=0).

0/0 key exercises

Polynomial graphs

Polynomial graphs and how the sign of the leading coefficient affects the broad shape of cubic and quartic graphs, and how real roots appear as x-intercepts where the curve touches or crosses the axis depending on whether the root is repeated or unique.

0/0 key exercises

Polynomial roots

A degree n polynomial has n roots counting multiplicity, and its roots may be real or complex; for a polynomial with real coefficients, complex roots occur in conjugate pairs, and the sum and product of roots are given by the coefficient relationships an−an−1 and an(−1)na0.