The Factor Theorem

We can use the factor theorem to find if a particular value ofis a root of a polynomial equation or to find out if a particular linear factor divides a polynomial perfectly, with no remainder.

The Factor Theorem

Ifis a factor of the polynomialthenhas no remainder or equivalently,

We show uses of the factor theorems in the following examples.

Example: Show thatis a factor of

Solveto get

thereforeis a factor of


Find the polynomial with with integer coefficients whose roots are

If these are the roots, the factors areororandhence we expand and simplifyto get


Whenis divided byorthe remainder is zero. Find a andhence find the polynomial

Remainder 0 whenis divided by

Remainder 0 whenis divided by

Now we solve the simultaneous equations




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