IM3 STEM - Chapter 8 Vocabulary

factor

(1) In arithmetic: When two or more integers are multiplied, each of the integers is a factor of the product. For example, 4 is a factor of 24 because 4 × 6 = 24. (2) In algebra: When two or more algebraic expressions are multiplied together, each of the expressions is a factor of the product. For example, x2 is a factor of -17x2y3 because (x2)( -17y3) = -17x2y3. (3) To factor an expression is to write it as a product. For example, the factored form of x2 - 3x - 18 is (x - 6)(x + 3). (4) A factor is part of a product. A polynomial expression p(x) is a factor of another polynomial expression P(x) when there is a polynomial q(x) such that p(x)q(x) = P(x). In the equation 3x2 - 9x + 6 = 3(x - 2)(x - 1), the expressions (x - 2), (x - 1) and 3 are factors.

root

(of a polynomial) The roots of a polynomial function P(x) are the values of x such that P(x) = 0. For example, the roots of P(x) = (x - 4)(2x + 3) are x = 4 and x = - , because these are the solutions to the equation (x - 4)(2x + 3) = 0. The x‑intercepts of a function's graph are the real roots of the function. Roots can be real or complex, but complex roots must be determined algebraically.

imaginary number

A number in the set of numbers created by multiplying the imaginary number i by every possible real number. The imaginary number i is defined as the square root of -1, or . Therefore i2 = -1. They are not positive, negative, or zero. These numbers are solutions of equations of the form x2 = (a negative number). In general, imaginary numbers follow the rules of real number arithmetic (e.g. i + i = 2i).

complex number

A number written in the form a + bi where a and b are real numbers. Each complex number has a real part, a, and an imaginary part, bi. Note that real numbers are also complex numbers where b = 0, and imaginary numbers are complex numbers where a = 0.

sum of cubes

A polynomial of the form x3 + y3. It can be factored as follows: x3 + y3 = (x + y)(x2 - xy + y2). For example, the sum of cubes 27y3 + 64 can be factored as (3y + 4)(9y2 - 12y + 16).

difference of cubes

A polynomial of the form x3 - y3. It can be factored as follows: x3 - y3 = (x - y)( x2 + xy + y2). For example, the difference of cubes 8x3 - 27 can be factored as (2x - 3)(4x2 + 6x + 9).

difference of squares

A polynomial that can be factored as the product of the sum and difference of two terms. The general pattern is x2 - y2 = (x + y)(x - y). For example, the difference of squares 4x2 - 9 can be factored as (2x + 3)(2x - 3).

double root

A root of a function that occurs exactly twice. If an expression of the form (x − a)2 is a factor of a polynomial, then the polynomial has a double root at x = a. The graph of the polynomial does not pass through the x-axis at x = a but is tangent to the axis at x = a.

polynomial

An algebraic expression that involves at most the operations of addition, subtraction, and multiplication. A polynomial in one variable is an expression that can be written as the sum of terms of the form: (any number) · (whole number). These polynomials are usually arranged with the powers of x in order, starting with the highest, left to right. The numbers that multiply the powers of x are called the coefficients of the polynomial. For example, x8 - 4x6 + 6x2 is a polynomial. Also see degree of a polynomial.

polynomial division

In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalised version of the familiar arithmetic technique called long division. It can be done easily by hand, because it separates an otherwise complex division problem into smaller ones.

real number

Irrational numbers together with rational numbers form the set of the real numbers. For example, the following are all real numbers: . All real numbers are represented on the number line.

coefficient

When variable(s) are multiplied by a number, the number is called a coefficient of that term. The numbers that are multiplied by the variables in the terms of a polynomial are called the coefficients of the polynomial. For example, 3 is the coefficient of 3x2.

zero (of a function)

See roots (of a polynomial function).

Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that every one-variable polynomial has at least one complex root (remember that every real number is also a complex number). This theorem can be used to show that every polynomial of degree n has n complex roots.

complex conjugate

The complex number a + bi has a complex conjugate a - bi. Similarly, the conjugate of c - di is c + di. What is noteworthy about complex conjugates is that both their product (a + bi)(a - bi) = a2 - b2i2 = a2 + b2 and their sum (a + bi) + (a - bi) = 2a are real numbers. If a complex number is a zero (or root) of a polynomial function with real coefficients, then its complex conjugate is also a zero (or root).

degree (of a polynomial)

The degree of a polynomial (in one variable) is the highest power of the variable. The degree of a polynomial function also indicates the maximum number of factors of the polynomial and provides information for predicting the number of "turns" the graph can take. The degree of a monomial is the sum of the exponents of its variables. For example, the degree of 3x2y5z is 8. For a polynomial, the degree is the degree of the monomial term with the highest degree. Example: for the polynomial 2x2y2 - 4x4z6 + x7z, the degree is 10.

x-intercept

The point where a graph intersects the x-axis. A graph may have several x-intercepts, no x-intercepts, or just one. In two dimensions, the coordinates of the x-intercept are (x, 0). In three dimensions they are (x, 0, 0). See intercepts.

quotient

The result of a division problem is a quotient with a remainder (which could be 0). When a polynomial P(x) is divided by a polynomial D(x), a polynomial Q(x) will be the quotient with a remainder R(x). The product of Q(x) and D(x) plus the remainder R(x) will equal the original polynomial. P(x) = D(x) · Q(x) + R(x).

remainder

When dividing polynomials in one variable, the remainder is what is left after the constant term of the quotient has been determined. The degree of the remainder must be less than the degree of the divisor. In the example below, the remainder is 3x - 3. . Also see quotient.