Third Root of Equation

Two of the roots of the equation are and . Find the third root of the equation.
Source: NCTM Mathematics Teacher, August 2006

Solution
Let be the third root of the equation. We write the equation in factored form

Multiply the right-hand side




Compare the coefficients of

Answer:

Alternative solution 1
Consider a general polynomial of third degree with roots and coefficients

Multiplying out the product on the right-hand side yields

provided that
Divide by

Third root =

Alternative solution 2
Substitute the value of into the equation



Substitute the value of into the equation



Multiply Eq. by and add to Eq.


——————–


Substitute the value of into Eq.


We now have the cubic equation  with two known roots . We can find the third root either by long division or by synthetic division: .
We can also graph the equation to find the third root

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