Factorise.
a⁶ + 5a² + 8
Do it quickly. Urgent.
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Factorise.
a⁶ + 5a² + 8
Do it quickly. Urgent.
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Over Reals
The range of the function is , so the graph doesn't meet the x-axis above it.
Hence, no point of intersection or no real zero.
It is irreducible over reals.
Over Complex
Let's factorize in a cubic polynomial.
Let .
This is a depressed cubic, which quadratic coefficient is 0.
Let's attempt to find the solutions via Cardano's method.
Cardano's Method
Two main ideas are
The second identity is a well-known one.
It can explain that one of the solutions is , then the equation we are solving is .
Hence, solving the depressed cubic means finding and such that
We deduce
Now, we can find the quadratic equation, which roots are and .
We can solve .
However, the solutions for are the perfect cube of and .
(Let and for convenience.)
Since is a real number we find pairs of and via , .
As , our solutions are , , .
Application
Now we can use this to solve our problem.
Given:
General depressed cubic form:
Quadratic equation:
According to the method mentioned above, the solutions are , , or . / and are two solutions for .
Hence our solutions for are
Answer
After we substitute back in, the six factors are