Review: Sequences, Polynomials, and Theorems - Vocabulary Flashcards

A set of vocabulary flashcards covering key terms from sequences, polynomials, and related theorems based on the lecture notes.

Arithmetic sequence

A sequence of numbers in which the difference between consecutive terms is constant (the common difference).

Common difference

The fixed amount added to obtain the next term in an arithmetic sequence.

Geometric sequence

A sequence in which the ratio of any term to the previous term is constant (the common ratio).

Common ratio

The fixed factor by which we multiply to get from one term to the next in a geometric sequence.

Arithmetic progression (AP)

Another term for an arithmetic sequence; a sequence with a constant difference between successive terms.

Geometric mean

For two numbers a and b, the geometric mean is √(ab); it is the middle term in a geometric progression between a and b.

Synthetic division

A shortcut method for dividing a polynomial by a binomial of the form x − c, using coefficients to obtain the quotient and remainder.

Quotient

The result of division of one polynomial by another (the part without the remainder).

Constant term

The term in a polynomial that does not contain the variable (x^0).

Degree of a polynomial

The highest power of the variable appearing in the polynomial.

Polynomial

An expression consisting of variables and coefficients combined by addition, subtraction, and multiplication.

Remainder

The amount left over after dividing one polynomial by another.

Remainder Theorem

When P(x) is divided by (x − r), the remainder equals P(r).

Factor Theorem

If P(r) = 0, then (x − r) is a factor of P(x).

Rational Root Theorem

Possible rational zeros of a polynomial are of the form p/q, where p divides the constant term and q divides the leading coefficient.

Factor

An expression that multiplies to give the polynomial; to express a polynomial as a product of its factors.

Divisor

A polynomial that divides another polynomial with no remainder.

Dividend

The polynomial being divided.

Infinite geometric series

A geometric series with infinitely many terms that converges if |r| < 1; its sum is a/(1 − r).

Sum of first n terms of a geometric sequence

Sn = a1(1 − r^n)/(1 − r) for r ≠ 1, giving the total of the first n terms.

Leading coefficient

The coefficient of the highest-degree term in a polynomial.

Geometric progression

Another term for a geometric sequence; each term is obtained by multiplying the previous term by a constant ratio.

Convergence condition for infinite geometric series

An infinite geometric series converges only if |r| < 1; its sum is a/(1 − r) when it converges.