SOLUTION: I don't understand why this is the answer to this problem. t^4 - 1 (t^2 + 1) (t + 1) (t - 1)

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Question 334833: I don't understand why this is the answer to this problem.
t^4 - 1
(t^2 + 1) (t + 1) (t - 1)
Found 3 solutions by Edwin McCravy, Fombitz, CharlesG2:
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!

Learn the rule for factoring the difference of two squares

Rule:  red%28A%29%5E2-green%28B%29%5E2=%28red%28A%29%2Bgreen%28B%29%29%28red%28A%29-green%28B%29%29

t%5E4+-+1

Write that as 

%28red%28t%5E2%29%29%5E2-green%281%29%5E2

Apply the rule above:

%28red%28t%5E2%29%2Bgreen%281%29%29%28red%28t%5E2%29-green%281%29%29

Write what's in the second parentheses this way:

%28t%5E2%2B1%29%28red%28t%29%5E2-green%281%29%5E2%29

Apply the rule to the second parentheses:

%28t%5E2%2B1%29%28red%28t%29%2Bgreen%281%29%29%28red%28t%29-green%281%29%29

(Notice that the rule does not work for the sum of two squares,
so we have to leave the first parentheses just as it is because
of the + between the terms.)

Edwin

Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Looking at the expression,t%5E4-1, it's the difference of two squares which can be factor as,
t%5E4-1=%28t%5E2-1%29%28t%5E2%2B1%29
The first expression on the right hand side is also a difference of two squares and can be further factored,
t%5E2-1=%28t-1%29%28t%2B1%29
When you put it all together,
t%5E4-1=%28t-1%29%28t%2B1%29%28t%5E2%2B1%29

Answer by CharlesG2(834) About Me  (Show Source):
You can put this solution on YOUR website!
"I don't understand why this is the answer to this problem.
t^4 - 1
(t^2 + 1) (t + 1) (t - 1)"

distributive property: a * (b + c) = ab + ac

example: 2 * (3 + 4) = 2 * 3 + 2 * 4 = 6 + 8 = 14

FOIL or First-Outer-Inner-Last is also distributive:

example: (ax + b)(cx + d) = ax * cx + ax * d + b * cx + b * d
(ax + b)(cx + d) = acx^2 + (ad + bc) * x + bd

associative property: this is just grouping, a + (b + c) = (a + b) + c
or a(bc) = (ab)c or 2(3*4) = (2*3)4

commutative property: this means you can switch the numbers you multiplying or adding around, a + b = b + a or ab = ba or 2*3 = 3*2

now back to the problem

t^4 - 1 (we want to factor this)

(t^2 + 1)(t^2 - 1)
test by FOIL: t^4 - t^2 + t^2 - 1 = t^4 - 1 (notice outer and inner terms cancel out)

t^2 - 1 can also be factored by same method
(t + 1)(t - 1)

t^2 + 1 can not be easily factored, well not unless you want to go into complex numbers (complex number is number of form a+bi where a and b are real numbers and i is the square root of -1)

if you do want to go into complex numbers:

t^2 + 1 = (t + i)(t - i) = t^2 - i^2 = t^2 - -1 = t^2 + 1

so without going into that:

t^4 - 1 = (t^2 + 1)(t + 1)(t - 1)

I hope this helps