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Which of the following are the factors of t^4 -81

2 Answers

1 vote

Answer:

±3

Explanation:

t^4 - 81 = 0

(t^2 -9) (t^2 + 9) = 0

t^2 = 9 = ±3

t^2 = -9 no solution

User Anoushka
by
8.2k points
2 votes

Answer:

I have a lot of factors mentioned at the end of this explanation.

Explanation:


t^4-81 is a difference of squares since we can write it as
(t^2)^2-(9)^2.

A difference of squares,
a^2-b^2, can be factored as
(a-b)(a+b).


t^4-81


(t^2)^2-(9)^2


(t^2-9)(t^2+9)

We see another difference of squares in this factored form.

I'm speaking of
t^2-9.

This can be rewritten as
(t)^2-(3)^2.

Let's factor it now.


t^2-9


(t)^2-(3)^2


(t-3)(t+3)

So
t^4-81=(t^2-9)(t^2+9)=(t-3)(t+3)(t^2+9).

Now
t^2+9 can also be factored if you invite all complex numbers into play.

You will need
i^2=-1 here.


t^2+9


t^2-(-9)


t^2-(9i^2)


t^2-(3i)^2

Now it is a difference of squares and we can do as we have been doing with the other factors:


(t-3i)(t+3i)

So the complete factored form of
t^4-81 is:


(t-3)(t+3)(t-3i)(t+3i).

So here are some things that you could say is a factor of
t^4-81:


1


-1


t-3


-(t-3)


-t+3 (same as one before; just a rewrite)


t+3


-(t+3)


-t-3 (same as one before; just a rewrite)


t^2-9


-(t^2-9)


-t^2+9 (same as one before; just a rewrite)


t^2+9


-(t^2+9)


-t^2-9 (same as one before; just a rewrite)


t-3i


-(t-3i)


-t+3i (same as one before; just a rewrite)


t+3i


-(t+3i)


-t-3i (same as one before; just a rewrite)

There are other ways to write some of these hopefully you can catch them on your own in your choice if they so occur.

User Regie
by
9.1k points

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