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The Rational Roots Test: Examples (page 2 of 2)

• Find all possible rational x-intercepts of y = 2x³ + 3x – 5.

Remember that x-intercepts are zeroes, and use the Rational Roots Test.

The constant term is 5, with factors 1 and 5.

The leading coefficient is 2, with factors 1 and 2.

Then the Rational Roots Tests yields the following possible solutions:

Don't forget the "plus-or-minus" on the solution. You either need to list out all the possible solutions separately, as I did in the first example, or use a "plus-or-minus" in front of each possible solution, as I showed here, or put one "plus-or-minus" in front of the whole list of possible solutions (as I will show in the next example). Just make sure you have  "plus-or-minus" in there somewhere.

• Use the Rational Roots Test to find all possible rational zeroes of
       6x⁴ – 11x³ + 8x² – 33x – 30.

This problem will be more complicated than the previous one, because the leading coefficient is not a simple "1". Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

The constant term is 30, with factors 1, 2, 3, 5, 6, 10, 15, and 30.

The leading coefficient is 6, with factors 1, 2, 3, and 6.

Then the Rational Roots Test yields:

Yes, this was a very long list. You should expect at least some of your homework or test questions to be as long as this.

You can see from the graph that we may have rational roots at x = – ²/₃ and x = + ⁵/₂, but it would not make sense to try any other of the listed potential roots.

In those last two examples, note how I was orderly in listing out the fractions, taking the time to reduce each fraction and to discard duplicates from the list. Take the time to work orderly like this, because this really is a simple topic, and it would be a shame to lose points only because you were careless.

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