Rational Root Theorem Examples

Find the possible rational roots of y = x⁵ + 3x² − 10x – 24.

To use the Rational Root Theorem, we need all of the possible factors, positive and negative, from our leading and lagging coefficients. Our leading factors are ±1. The factors for the last term are more complicated: ±1, ±2, ±3, ±4, ±6, ±8, ±12, and ±24.

Whew, what a pain. Now we take the second set (from the constant term) and put each and every one on top of each and every one from the first set. It seems a little backwards, but that's how we do it. Constant over leading term.

In our case, that's easy, because the leading terms are ±1. So, our final list of possible factors is: ±1, ±2, ±3, ±4, ±6, ±8, ±12, and ±24.

Find the possible rational roots of y = 3x⁷ − 12x³ + 52x² − 96x + 9.

Our constant factors are ±1, ±3, ±9. These are the ones that go on top. The leading factors are ±1, ±3. We'll stick these below. Let's write all the combinations out:

We can have any combination of positive and negative numbers our hearts desire, so these are all ± as well.

Our final answer is ±1, ±3, ±9, and ±.

Find all the roots of y = x⁴ − 5x² + 4.

A quick application of the Rational Root Test gives us the following possible roots: ±1, ±2, and ±4. The leading 1 makes this simple.

When checking roots, it's usually a good idea to start with 1; it's always there when we need it, and it is easy to plug in. What a good friend 1 is.

f(1) = 1 – 5 + 4 = 0

Bingo bango, got it in 1. Let's factor it by using synthetic division:

Our polynomial is now y = (x – 1)(x³ + x² − 4x – 4). If we wanted to, we could use the Rational Root Theorem on our new degree 3 polynomial, find a root for it, and try factoring it that way. We see another way, though: factoring by grouping.

x²(x + 1) – 4(x + 1)

= (x + 1)(x² – 4)

= (x + 1)(x + 2)(x – 2)

That worked better than expected, because we remembered the difference of two squares. Remembering things: we here at Shmoop highly recommend it.

Now we tie it all together. Our original polynomial, y = x⁴ − 5x² + 4, has become:

y = (x – 1)(x + 1)(x + 2)(x − 2)

Our roots are x = 1, -1, 2, and -2. Most of our possible roots were actually roots. That doesn't happen very often.