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)#