Polynomials, Cubic Equations and Partial Fractions

These flashcards cover key terms and concepts related to polynomials, cubic equations, and partial fractions, including definitions, important theorems, and methods.

Polynomial

An algebraic expression of the form an x^n + an−1 x^(n−1) + … + a2 x^2 + a1 x + a0, where n is a non-negative integer and a0, a1, a2,…, an are constants.

Coefficient

The constant 'a' in the term a*x^n, representing the multiplicative factor of that term.

Degree of a polynomial

The highest power of x in the polynomial.

Constant term

The term in the polynomial that does not contain any variables, often referred to as the coefficient of x^0.

Evaluating polynomials

Finding the value of the polynomial P(x) by substituting a specific value for x, denoted as P(a).

Adding polynomials

Combining polynomials by summing their like terms.

Multiplying polynomials

Finding the product of polynomials using the Distributive Law and combining like terms.

Remainder Theorem

States that the remainder of the division of a polynomial f(x) by (x-a) is f(a).

Factor Theorem

If f(a) = 0, then (x - a) is a factor of the polynomial f(x).

Synthetic Division

A simplified method of dividing polynomials, particularly useful for finding roots and coefficients.

Cubic equation

An equation of the form ax^3 + bx^2 + cx + d = 0, where a, b, c, and d are constants.

Identity in polynomials

An equality that holds true for all values of x, established by equal coefficients of corresponding powers.

Long Division for Polynomials

Method used to divide a polynomial by another polynomial, similar to arithmetic long division.

Algebraic fraction

A fraction where both the numerator and the denominator are polynomials.

Proper algebraic fraction

An algebraic fraction where the degree of the numerator is less than the degree of the denominator.

Improper algebraic fraction

An algebraic fraction where the degree of the numerator is greater than or equal to the degree of the denominator.

Partial fractions

Process of breaking down a complex rational expression into simpler fractions that are easier to work with.

Roots of cubic equations

Solutions of cubic equations, which can be real or complex and can be found using various methods, including factorization.