Answer to Question #191564 in Algebra for EUGINE HAWEZA

Question

i. Write notes on the remainder and factor theorem (polynomials).

ii. Write short notes on Indices and Logarithms (highlighting the rules and siting examples of their mathematical application)

Answer 1

(i)Remainder Theorem :

Let f(x) be a polynomial,

Let its divisor (x-h) where h is a positive integer,

Let it's quotient be q(x),

Let its reminder be r,

So, $f(x) = (x-h).q(x) + r$

But,

If we substitute h in the place of x,

Then, $f(h) = (h-h).q(h) + r$

So, $f(h) = r$

Therefore, If we put h in the place of x in f(x), then we will get our remainder.

Factor Theorem :

It states that if f(h) = 0, then (x-h) is a factor of f(x).

(ii) Index (indices):

 In Mathematics, indices is the power or exponent which is raised to a number or a variable

For example, in number $2^4$ , 4 is the index of 2. The plural form of index is indices.

Logarithms:

In mathematics, the logarithm is the inverse function to the exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x.

In our simplest case, the logarithm counts the number of occurrences of the same factor in repeated multiplication; e.g., since$\ 1000 = 10 × 10 × 10 = 10^3$ , the "logarithm base 10" of 1000 is 3, or $log_{10}(1000) = 3$ .