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Chapter 7: Functions

In a linear equation, if one variable varies, the other one also changes. Two variables x and y are said to vary directly if and only if there is a constant of variation, k. k ¹ 0, such that y = kx. If x ¹ 0, y/x = k.

There is a special kind of linear equation called the step function. The step function is represented in the form of [x], which means the greatest integer that is equal or smaller than x. The step function is especially useful in accounting and statistic.

# Polynomial Function

We have discussed about polynomials in the previous chapters. The degree of a polynomial in one variable is the degree of the term of the highest degree when the polynomial is written in standard form. The leading coefficient of the polynomial is the coefficient of the term with the highest degree. A polynomial function is a function P whose values are defined by a polynomial. P(x) = anxn + a n-1 x n+1 + … + a1x + ao. Here are some additional properties of the polynomial functions:

The Remainder Theorem: For each polynomial P(x) of degree n ³ 1 and each real number r, there is a polynomial Q(x) of degree n - 1, such that P(x) = (x - r)Q(x) + P(r).

The Factor Theorem: For each polynomial P(x), of degree n ³ I and each real number r, (x - r) is a factor of P(x) if and only if P(r) = 0.

The Fundamental Theorem of Algebra: Each nth-degree polynomial, n ³ 1, can be factored into n linear factors.

or

Every polynomial equation of degree n has exactly n roots, where a root of multiplicity k is counted as k roots.

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