Editing Rational root theorem (section)

==Application==

The theorem is used to find all rational roots of a polynomial, if any. It gives a finite number of possible fractions which can be checked to see if they are roots. If a rational root {{math|1=''x'' = ''r''}} is found, a linear polynomial {{math|(''x'' – ''r'')}} can be factored out of the polynomial using [[polynomial long division]], resulting in a polynomial of lower degree whose roots are also roots of the original polynomial.

===Cubic equation===

The general [[cubic equation]]
<math display="block">ax^3 + bx^2 + cx + d = 0</math>
with integer coefficients has three solutions in the [[complex plane]]. If the rational root test finds no rational solutions, then the only way to express the solutions [[algebraic expression|algebraically]] uses [[Cubic function|cube roots]]. But if the test finds a rational solution {{math|''r''}}, then factoring out {{math|(''x'' – ''r'')}} leaves a [[quadratic polynomial]] whose two roots, found with the [[quadratic formula]], are the remaining two roots of the cubic, avoiding cube roots.