# How do you factor #r^4 + r^3 − 3r^2 − 5r − 2#?

##### 1 Answer

#### Explanation:

One way to do this is finding one root of the term and then performing polynomial long division. The procedure can be repeated until there is only a quadratic term left.

**1) Searching for the first root / factor**

If searching for a root, it is generally a good idea to evaluate the term for values like

Here, it works with

#(-1)^4 + (-1)^3 - 3 (-1)^2 - 5 * (-1) - 2 = 1 - 1 - 3 + 5 - 2 = 0#

Thus,

**2) Polynomial long division**

Let's use divide by

#color(white)(xx)(r^4 + r^3 - 3r^2 - 5r - 2) -: (r + 1) = r^3 - 3r - 2#

# -(r^4 + r^3)#

#color(white)(x) color(white)(xxxxxx)/#

#color(white)(xxxxxx)0 - 3 r^2 - 5r#

#color(white)(xxxx)-( - 3 r^2 -3r)#

#color(white)(xxxxxx) color(white)(xxxxxxxxx)/#

#color(white)(xxxxxxxxxxx)-2r - 2#

#color(white)(xxxxxxxxx)-(-2r - 2)#

#color(white)(xxxxxxxxxxx) color(white)(xxxxxxxx)/#

#color(white)(xxxxxxxxxxxxxxxxx)0#

Thus, you already can factor your term as follows:

#r^4 + r^3 - 3r^2 - 5r - 2 = (r+1)(r^3 - 3r -2)#

**3) Searching for the second root / factor**

Now, let's try and factor

We can repeat the same procedure as before and again, we find that

#(-1)^3 - 3* (-1) - 2 = -1 + 3 - 2 = 0#

**4) Polynomial long division**

Thus, we can divide

# color(white)(xx) (r^3 color(white)(xxxx) - 3r - 2) -: (r+1) = r^2 - r - 2#

# - (r^3 + r^2)#

# color(white)(x) color(white)(xxxxxx) /#

# color(white)(xxxx) -r^2 - 3r#

# color(white)(xx) -(-r^2 - r)#

# color(white)(xxxx) color(white)(xxxxxxxx) /#

# color(white)(xxxxxxxx) -2r - 2#

# color(white)(xxxxxx) -(-2r - 2)#

# color(white)(xxxxxxxx) color(white)(xxxxxxxx) /#

# color(white)(xxxxxxxxxxxxxx) 0#

At this point, we can factor the term as follows:

#r^4 + r^3 - 3r^2 - 5r - 2 = (r+1)(r+1)(r^2 - r - 2)#

**5) Factoring the quadratic term**

At this point, the only thing left to do is factoring the term

There are a lot of ways to do that. Let me show you one of my favourites.

Basically, you would like to have something like this:

#r^2 - r - 2 = (r +a)(r+b)#

# = r^2 + (a + b)r + a * b#

Thus, you need to find

The solution to this is

Thus, your quadratic term can be factored as follows:

#r^2 - r - 2 = (r + 1)(r - 2)#

**6) Solution**

In total, you have found the following factorization:

#r^4 + r^3 - 3r^2 - 5r - 2 = (r+1)(r+1)(r+1)(r-2)#

# = (r+1)^3(r-2)#