What are the factors for #a^6 + 7a^3 + 6#?
A simple manipulation of your variables will give us the solution.
Let's kind of factor your function as
Now, we can rewrite it as
Solving it, we will find the roots:
These are our new factors. Now, we can factor the newfound equation using them:
Note that when you expand these factors you will go back to the original
Note, again, that if you expand it you will get exactly
For the sake of a more full answer:
"Factoring completely" without context is ambiguous.
What we need to remove the ambiguity is a statement of what kinds of coefficients we are using.
The integers are the "positive and negative whole numbers and
The rational numbers are the ratios (fractions) of integers.
The real numbers include all numbers that are not imaginary (that do not involve
The complex numbers are all two-part numbers
To factor over the integers (or over the rational numbers) means that we are going to use integer (or rationals) This is what beginning and intermediate algebra classes usually mean by "factor completely"..
As others have shown:
These polynomials cannot be factored using integer (or rational) coefficients.
We are finished if we are factoring over the integers (or the rationals)
If we are factoring over the real numbers, (This is what some more advanced algebra and precalculus classes mean by "factor completely".)
We observe that
The two quadratics have no factors over the real numbers.
Finally, we may be factoring over the complex numbers, as some parts of advanced algebra or precalculus classes sometimes do.we continue. (Often this is asked by including the phrase "over the complex numers" of "find all linear factors of"
We use the quadratic formula (or other methods) to find the zeros of the two quadratics. This gives us the linear factors over the complex numbers.
If you need to do this, let me know and we'll go through it.