Algebra and functions

Question 1

How can an algebraic fraction be simplified?

Answer 1

By searching for a common factor of both the numerator and the denominator, and then dividing both by this common factor

Question 2

How can one polynomial be divided by another polynomial using long division?

Answer 2

First transform the polynomials so that the terms are in descending order of their power and so there is a term for every power, i.e. transform 1+x² into x²+0x+1. Then, divide the first term of the dividend by the highest term of the divisor, and place the result above the bar. Multiply the divisor by the result just obtained, and subtract this from the appropriate terms of the original dividend. Then, “pull down” the next term from the dividend. Repeat these steps, except use the terms that have just been written as the dividend. When there is nothing to “pull down”, the remaining expression is the remainder

Question 3

What does the factor theorem state?

Answer 3

A polynomial f(x) has factor (x-k) if and only if f(k)=0

Question 4

What does the remainder theorem state?

Answer 4

If a polynomial f(x) is divided by (ax-b), the remainder can be calculated by f(b/a)