Question #48cc8

1 Answer
Jul 7, 2015

How you find the factors depends a lot on what tools you have in your mathematical toolbox.


Note on terminology, although there is a "quartet" of terms, the word "quartic" refers to the degree being 4. Not to the number of terms. We could call it a quadrinomial, but I don't hear that or see it in textbooks.

To factor a cubic polynomial, First check for common factors and check for sum or difference of two cubes. (Not relevant in this problem.)

I would then try factoring by grouping . It won't always work, but it doesn't take long to try it:

#y= x^3 - 9x^2 + 27x - 27#

# (x^3 - 9x^2) + (27x - 27)#

#x^2(x-9) + 27(x-1)# -- No, factoring by grouping won't work.

You may have noticed that both the first and last terms are perfect cubes. If so, you might try using the guess #(x -3)^3#
Or, you might use the rational zero theorem to learn that #3# is a zero, so #x-3# is a factor. Then do the division to get:

#x^3-9x^2+27x-27 = (x-3)(x^2-6x+9)#

Which can be further factored to get:

#x^3-9x^2+27x-27 = (x-3)^3#