Here are three algebraic formulas, the binomial formulas, which can be used for factoring:
You should check that these formulas work by multiplying out using the FOIL method.
Here is how to use these formulas for factoring purposes: Let's say we want to factor
We can write this polynomial as
and then notice that the terms match the second binomial formula for the values a=2x and b=3. Consequently,
and we have factored the polynomial completely. Note that x=3/2 is the only root, with multiplicity 2.
Here is another example: factor the polynomial
We can write the polynomial as the difference of two squares and then use the third binomial formula:
As an example, let us factor the polynomial
We can rewrite this polynomial as
Now it matches formula (5) with a=2x and b=3. Consequently
The polynomial has a triple root at x=3/2.
Say, we like to factor . By formula (6), we can write
In this case the factorization is complete, since the polynomial is an irreducible quadratic polynomial.
What about the polynomial ? We first write this as the difference of two cubes, and then use formula (7):
Aside: Note that the factorization is still not complete. The Fundamental Theorem of Algebra tells us that it is possible to factor
further. Since you can see from the graph of this polynomial that it does not have real roots, the polynomial
can be factored into 2 irreducible quadratic polynomials. To find these two polynomials requires more familiarity with complex numbers; you can check that
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