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Please be aware! Xronos is very good at making sure the factored polynomial you
have entered is the same polynomial. It is less reliable to know if you have entered a
fully factored version of that polynomial. To be sure that your factored version is
fully factored, you should check two things:
Every factor is degree 1 or degree 2 polynomial (a linear expression or a
quadratic).
If the factor is a quadratic, then the discriminate () must be negative.
If it is Zero or positive, then that term can be factored further and you
should try to do so.
Note: This is using an experimental factoring validator. If you verified that your
answer should be correct and Xronos won’t take it, please email your instructor to
see if there is a problem.
1 :
Factor the following polynomial (hint: Using the Rational Root Theorem to get a
zero, then divide out the factor to reduce the polynomial)
Remember that the Rational Root Theorem says you want to find all the factors of
the constant (i.e. ) and all the factors of the leading coefficient (i.e. ) and
create a list of () every combination of a factor of divided by a factor of
.
Note that this is probably going to give you a long list! Unfortunately there really
isn’t a great way to know which of the factors to try first; so I would typically suggest
starting with the ones that are easiest to calculate... This is why the Rational Root
Theorem should always be a tool of last resort! It takes a long time, and it is really
inefficient, but sometimes it’s all you have. In this spirit, once you have found a zero
and factored it out using something like polynomial long division, make sure to start
over on the resulting polynomial in terms of factoring; with any luck you
won’t need to use RRT and can use some other technique (like grouping
or quadratic form). The goal is to use RRT as infrequently as possible!
2 :
Factor the following polynomial (hint: Using the Rational Root Theorem to get a
zero, then divide out the factor to reduce the polynomial)
Remember that the Rational Root Theorem says you want to find all the factors of
the constant (i.e. ) and all the factors of the leading coefficient (i.e. ) and
create a list of () every combination of a factor of divided by a factor of
.
Note that this is probably going to give you a long list! Unfortunately there really
isn’t a great way to know which of the factors to try first; so I would typically suggest
starting with the ones that are easiest to calculate... This is why the Rational Root
Theorem should always be a tool of last resort! It takes a long time, and it is really
inefficient, but sometimes it’s all you have. In this spirit, once you have found a zero
and factored it out using something like polynomial long division, make sure to start
over on the resulting polynomial in terms of factoring; with any luck you
won’t need to use RRT and can use some other technique (like grouping
or quadratic form). The goal is to use RRT as infrequently as possible!
3 :
Factor the following polynomial (hint: Using the Rational Root Theorem to get a
zero, then divide out the factor to reduce the polynomial)
Remember that the Rational Root Theorem says you want to find all the factors of
the constant (i.e. ) and all the factors of the leading coefficient (i.e. ) and
create a list of () every combination of a factor of divided by a factor of
.
Note that this is probably going to give you a long list! Unfortunately there really
isn’t a great way to know which of the factors to try first; so I would typically suggest
starting with the ones that are easiest to calculate... This is why the Rational Root
Theorem should always be a tool of last resort! It takes a long time, and it is really
inefficient, but sometimes it’s all you have. In this spirit, once you have found a zero
and factored it out using something like polynomial long division, make sure to start
over on the resulting polynomial in terms of factoring; with any luck you
won’t need to use RRT and can use some other technique (like grouping
or quadratic form). The goal is to use RRT as infrequently as possible!
4 :
Factor the following polynomial (hint: Using the Rational Root Theorem to get a
zero, then divide out the factor to reduce the polynomial)
Remember that the Rational Root Theorem says you want to find all the factors of
the constant (i.e. ) and all the factors of the leading coefficient (i.e. ) and
create a list of () every combination of a factor of divided by a factor of
.
Note that this is probably going to give you a long list! Unfortunately there really
isn’t a great way to know which of the factors to try first; so I would typically suggest
starting with the ones that are easiest to calculate... This is why the Rational Root
Theorem should always be a tool of last resort! It takes a long time, and it is really
inefficient, but sometimes it’s all you have. In this spirit, once you have found a zero
and factored it out using something like polynomial long division, make sure to start
over on the resulting polynomial in terms of factoring; with any luck you
won’t need to use RRT and can use some other technique (like grouping
or quadratic form). The goal is to use RRT as infrequently as possible!
