how to factor this binomial?
25x^2+1

Question

how to factor this binomial? 
25x^2+1

anonymous · Answer

(-i+5 x) (i+5 x)

asnaseer · Answer

$$a^2+b^2=(a+ib)(a-ib)$$

mertsj · Answer

That is the sum of two squares and it is prime within the set of real numbers.

mertsj · Answer

What class are you taking?

anonymous · Answer

algebra...

mertsj · Answer

So have you had imaginary numbers yet?

anonymous · Answer

no

mertsj · Answer

Then put down that this binomial cannot be factored and is prime.

mertsj · Answer

If it were the difference of two squares, it would be factorable but since it is the sum of two squares, it is prime.

anonymous · Answer

so what's the difference of sum/diff of squares?

mertsj · Answer

x^2 is a square
16 is a square
x^+16 is the sum of two squares and cannot be factored
x^2-16 is the difference of two squares and can be factored.

mertsj · Answer

Sum means add.
Difference means subtract.

anonymous · Answer

so whenever there is a - sign it can be factored and when + is prime?

mertsj · Answer

Yes.  If those signs join two squares.

asnaseer · Answer

Mertsj: why do say this must be prime? (e.g. when x=1, we get $25x^2+1=26$ which is not a prime number.

mertsj · Answer

Prime means has no factors except itself and 1

asnaseer · Answer

so mean the "expression" is prime rather than the number that it produces?

mertsj · Answer

Yes

asnaseer · Answer

ok - thx for clarifying

mertsj · Answer

5 is prime

mertsj · Answer

x^2+4 is prime

mertsj · Answer

yw

anonymous · Answer

got you!!!

mertsj · Answer

Good

directrix · Answer

x^2 + 4 is a prime expression over the set of Reals but is not prime over the set of Complex numbers.