# How do you factor #x^3+6x^2+11x+6#?

##### 3 Answers

#### Answer:

#### Explanation:

There are several ways to approach this.

One of the most reliable is to hope that the expression has rational roots and apply the Rational Root Theorem.

In this case, the Rational Root Theorem tells us that (if the expression has rational roots) those roots are integer factors of

We can build a table/spreadsheet to evaluate the expression for these possible factors:

Note that when

Remember that if an expression has a value of

then

Therefore, we have three factors for the given expression:

Since the given expression is of degree

#### Answer:

#### Explanation:

Given:

#x^3+6x^2+11x+6#

You are actually likely to encounter this pattern of coefficients

For illustration purposes, let us see what happens if we start by applying a standard method used when solving a general cubic:

**Tschirnhaus transformation**

In order to simplify any cubic of the form

In our example,

Note that:

#(x+2)^3 = x^3+6x^2+12x+8#

So we find:

#x^3+6x^2+11x+6 = (x+2)^3-(x+2)#

#color(white)(x^3+6x^2+11x+6) = t^3-t#

#color(white)(x^3+6x^2+11x+6) = t(t^2-1)#

#color(white)(x^3+6x^2+11x+6) = t(t-1)(t+1)#

#color(white)(x^3+6x^2+11x+6) = (x+2)(x+1)(x+3)#

In more sensible order:

#x^3+6x^2+11x+6 = (x+1)(x+2)(x+3)#

#### Answer:

#### Explanation:

It's probably worth mentioning that there's another way to solve this factoring problem if you have a calculator handy.

Given:

#x^3+6x^2+11x+6#

Since all of the coefficients of this polynomial are positive, we get an easy to calculate number if we put

#color(red)(1)(color(blue)(100))^3+color(purple)(6)(color(blue)(100))^2+color(brown)(11)(color(blue)(100))+color(green)(6) = color(red)(1)color(purple)(06)color(brown)(11)color(green)(06)#

That is, the number formed by writing two digits for each coefficient (apart from the leading one, which does not need more than

Then factors of the form

So we can try dividing

The rational root theorem can be used before that to let us know that the only possible rational zeros are:

#+-1, +-2, +-3, +-6#

Hence we want to try factors:

In fact we find:

#1061106 / 101 = 10506#

and:

#10506/102 = 103#

Hence we can deduce that:

#x^3+6x^2+11x+6 = (x+1)(x+2)(x+3)#