How do you find all rational zeroes of the function using synthetic division [Math Processing Error]?

1 Answer
Apr 16, 2017

The "possible" rational zeros are [Math Processing Error]

The actual zeros are: [Math Processing Error], [Math Processing Error] and [Math Processing Error]

Explanation:

Given:

[Math Processing Error]

First note that all of the coefficients are divisible by [Math Processing Error], so separate that out as a factor...

[Math Processing Error]

By the rational roots theorem, any rational zeros of [Math Processing Error] are expressible in the form [Math Processing Error] for integers [Math Processing Error] with [Math Processing Error] a divisor of the constant term [Math Processing Error] and [Math Processing Error] a divisor of the coefficient [Math Processing Error] of the leading term.

That means that the only possible rational zeros are:

[Math Processing Error]

In addition note that the sum of the coefficients is [Math Processing Error]. That is:

[Math Processing Error]

So [Math Processing Error] is a zero and [Math Processing Error] a factor.

We can use synthetic division to find:

[Math Processing Error]

It looks something like this:

[Math Processing Error]
[Math Processing Error]
[Math Processing Error]

where the final [Math Processing Error] shows us that the remainder is [Math Processing Error] as expected.

To factor the remaining quadratic, note that [Math Processing Error] and [Math Processing Error], so:

[Math Processing Error]

So:

[Math Processing Error]

with zeros [Math Processing Error], [Math Processing Error] and [Math Processing Error].