Here’s a challenge!
Solve the following cubic equation, and discover that the real-number root is an integer:
(31/2x− 12 1/2)^3 = 0
Solution
Remember that the 1/2 power of a number is the positive square root. The above equation is therefore not
ambiguous. It’s a legitimate cubic equation in x, because the 1/2 powers involve only the coefficient and
the constant, not the variable x itself. Both the coefficient and the stand-alone constant are irrational, so
this equation looks difficult! But for the moment, let’s forget about that. We can solve it using the above
general formula. We have a= 3 1/2 and b=−(121/2). Therefore
x=−b/a
=−[−(121/2)] / 31/2
= 12 1/2 / 31/2
Using the power of quotient rule from Chap. 9, we can simplify this root and then reduce it to an integer:
x= 12 1/2 / 31/2
= (12/3)1/2
= 4 1/2
= 2
We should check to be sure that this root really works in the original equation! Let’s plug it in and
find out:
(31/2x− 12 1/2)^3 = 0
(31/2× 2 − 12 1/2)^3 = 0
(31/2× 4 1/2− 12 1/2)^3 = 0
Using the power of product rule from Chap. 9, we can simplify this to
[(3× 4)1/2− 12 1/2]^3 = 0
(121/2− 12 1/2)^3 = 0
03 = 0
0 = 0
Three Binomial Factors
When a cubic equation can be expressed as three binomial factors, those factors are rarely all
the same. Two of them might be identical, but often all three are different. Cubics of this sort
are in binomial-factor form.
Three Binomial Factors 415
