Question

Jay factored the 4 term polynomial x^3-9x\:+\:2x^2-18 x 3 − 9 x + 2 x 2 − 18 and decided that the complete factorization was (x+2\right) ( x + 2 ) \left(x^2-9\right) ( x 2 − 9 ) .

Before turning in his paper, he checked his final factorization by multiplying out his factors and was sure that he had found the correct factors. When his teacher graded his paper, she marked his answer as incorrect, but gave him one more chance to show the correct factorization.

a) What was Jay's mistake?



b) Show/describe EACH step in factoring the 4 term expression correctly and completely:

Answer 1

Correct factorization: (x+2)(x+3)(x-3)

Explanation:

The given 4 term polynomial is:

`x^(3)-9x+2x^(2)-18`

Part a) Jay's Mistake:

Factorization of Jay was:

`(x+2)(x^(2)-9)`

Though this expression will simplify to original given expression but this is not the complete and final factorization. The second factor which is x² - 9 can be factored further, which is shown in the next part.

Part b) Complete Factorization

In order to factor a 4 term expression of the type given in the question, the first step is to take the common from similar terms. You might need to re-arrange the terms before taking common in some case. Taking commons from the given expression, we get:

`x^(3)-9x+2x^(2)-18\\\\ = x(x^(2)-9)+2(x^(2)-9)\\\\ =(x+2)(x^(2)-9)`

Jay stopped at this step. At this step you need to look if any part of the expression can be factored further. Luckily, in this case x² - 9 can be factored further as its a difference of perfect squares:

x² - 9 = x² - (3)² = (x + 3)(x - 3)

Using these factors of x² - 9 in previous expression, we get:

`(x+2)(x^(2)-9)\\\\ = (x+2)(x+3)(x-3)`

This is the final factored form of the given 4 term expression as it can not be factored further in any way.