# Use of the Factor Theorem for Rational Values of the Variable for Polynomials

## Use of the Factor Theorem for Rational Values of the Variable for Polynomials

# The Factor Theorem

## Understanding the Factor Theorem

- The
**Factor Theorem**is a powerful tool in algebra which allows us to determine if a given expression is a factor of a polynomial. - Essentially, the theorem states that a polynomial
`f(x)`

has a factor`(x-a)`

if and only if`f(a) = 0`

.

## Applying the Factor Theorem

- To apply the Factor Theorem, plug in the value of
`x`

from the possible factor`(x-a)`

into the polynomial. - If the answer is zero, then
`(x-a)`

is a factor of the polynomial.

## Factor Theorem and Division

- The connection to division is that if
`(x-a)`

is a factor of the polynomial`f(x)`

, when`f(x)`

is divided by`(x-a)`

, the remainder will be zero. - One can therefore think of the Factor Theorem as a quick way of performing polynomial division.

## Factor Theorem and Roots

- Note that if
`f(a) = 0`

for some value of`a`

, then`a`

is called a**root or a zero**of the polynomial. - Roots are the values of
`x`

for which the value of the polynomial is zero. - There exists a direct relationship between the roots of a polynomial, and the factors of a polynomial.

## Synthetic Division

- Synthetic division is a shorthand method of performing long polynomial division. It’s notably used with the Factor theorem.
- In
**synthetic division**, coefficients are used instead of variables, which simplifies the division process significantly.

Most importantly, the Factor Theorem provides a simple and effective strategy to factorise polynomials, especially those of higher degree. Just as importantly, it gives clues about the graph of the polynomial: wherever the polynomial touches or crosses the `x`

axis, `(x-a)`

is a factor, and `a`

is a root.