Factoring: Some Special Cases

The binomial formulas.

Here are three algebraic formulas, the binomial formulas, which can be used for factoring:

eqnarray11

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

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We can write this polynomial as

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and then notice that the terms match the second binomial formula for the values a=2x and b=3. Consequently,

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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

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We can write the polynomial as the difference of two squares and then use the third binomial formula:

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The cube of a binomial.

There are similar formulas to factor some special cubic polynomials:

eqnarray23

As an example, let us factor the polynomial

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We can rewrite this polynomial as

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Now it matches formula (5) with a=2x and b=3. Consequently

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The polynomial has a triple root at x=3/2.


Sums and differences of two cubes.

here are two more formulas to handle special cases of cubic polynomials:

eqnarray26

Say, we like to factor tex2html_wrap_inline142 . By formula (6), we can write

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In this case the factorization is complete, since the polynomial tex2html_wrap_inline144 is an irreducible quadratic polynomial.

What about the polynomial tex2html_wrap_inline146 ? We first write this as the difference of two cubes, and then use formula (7):

eqnarray29

Aside: Note that the factorization is still not complete. The Fundamental Theorem of Algebra tells us that it is possible to factor

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further. Since you can see from the graph of this polynomial that it does not have real roots, the polynomial

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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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Exercise 1.

Factor the polynomial tex2html_wrap_inline148 completely.

Answer.

Exercise 2.

Factor the polynomial tex2html_wrap_inline150 completely.

Answer.

Exercise 3.

Factor the polynomial tex2html_wrap_inline152 completely.

Answer.

Exercise 4.

Factor the polynomial tex2html_wrap_inline154 completely.

Answer.

Exercise 5.

Factor the polynomial tex2html_wrap_inline156 completely.

Answer.

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