We can use the factor theorem to find if a particular value of
is 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
If
is a factor of the polynomial
then
has no remainder or equivalently,
We show uses of the factor theorems in the following examples.
Example: Show that
is a factor of
Solve
to get
thereforeis a factor of
Example
Find the polynomial with with integer coefficients whose roots are
If these are the roots, the factors are
or
or
and
hence we expand and simplify
to get
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Example
When
is divided by
or
the remainder is zero. Find a and
hence find the polynomial
Remainder 0 whenis divided by
Remainder 0 whenis divided by
Now we solve the simultaneous equations
(1)
(2)
hence