Exercises - using factor theorem
- Find the remainder when
is divided by
.
- Use the factor theorem to factorise
completely.
-
- Find
.
- Factorise
completely
- Use the Factor Theorem to determine all the factors of the following expression:
- Complete: If
is a polynomial and
is a number such that
, then
is .....
Solving cubic equations
Once you know how to factorise cubic polynomials, it is also easy to solve cubic equations of the kind
Solve
-
Try
Therefore
is NOT a factor.
Try
Therefore
IS a factor.
-
The first term in the second bracket must be
to give
if one works backwards.
The last term in the second bracket must be
because
.
So we have
.
Now, we must find the coefficient of the middle term (
).
gives
. So, the coefficient of the
-term must be 7.
So,
.
-
can be further factorised to
,
and we are now left with
-
Sometimes it is not possible to factorise the trinomial ("second bracket"). This is when the quadratic formula
can be used to solve the cubic equation fully.
For example:
Solve for
:
-
Try
Therefore
is NOT a factor.
Try
Therefore
is NOT a factor.
Therefore
IS a factor.
-
The first term in the second bracket must be
to give
.
The last term in the second bracket must be 2 because
.
So we have
.
Now, we must find the coefficient of the middle term (
).
gives
. So, the coefficient of the
-term must be
. (
)
So
.
cannot be factorised any futher and we are now left with
-
-
Always write down the formula first and then substitute the values of
and
.
-
Exercises - solving of cubic equations
- Solve for
:
- Solve for
:
- Solve for
:
- Solve for
:
:
Remove brackets and write as an equation equal to zero.
- Solve for
if
End of chapter exercises
- Solve for
:
-
- Show that
is a factor of
- Hence, by factorising completely, solve the equation
-
for all values of
. What is the value of
?
-
- Use the factor theorem to solve the following equation for
:
- Hence, or otherwise, solve for
:
-
A challenge :
Determine the values of
for which the function
leaves a remainder of 9 when it is divided by
.