Question

In your own words, explain the remainder theorem and give an example of how it is used. How is the factor theorem related to the remainder theorem? What is synthetic division? Demonstrate how synthetic division is used to find zeros of a polynomial.

Answer 1

if we divide a polynomial f(x) by a factor ( x- a) we get remainder f(a)

for example :

2x^2 - 5x -1 divided by x - 3

we get remainder = 2

according to the factor theorem

if (x - a) is a factor of polynomial f(x)

then it means on dividing remainder is 0

synthetic division is a shortcut method of dividing polynomials by a linear factor

lets take an example

to find the zeros of this polynomial

f(x) = x^2 - 5x + 6

we first divide the polynomial by x = 2

on dividing we get remainder is 0

this means x = 2 is one of the zero and from the result

x - 3 = 0

which means other zero is

x = 3

so, two zeros are

x = 2 , 3