Question

Fill in the missing numbers to complete the factorization. Some of the numbers could be negative. Type the numbers in increasing order.

x 3 + 2x 2 - x - 2 =
(x= ?)
(x= ?)
(x= ?)

Answer 1

apply the factor and rational roots theorems
because of the last term -2 then -2 ,2, 1, or -1 maybe roots of f(x) = 0

what does f(-2) work out to if its 0 then by the factor theorem (x + 2) will be a factor.

Answer 2

f(-2) = (-2)^3 + 2(-2)^2 - (-2) - 2 = ?

Answer 3

can you do the above calculation?

Answer 4

I agree with welshfella that one or more possible roots of this polynomial could be -2, 2, -1 and/or 1. How would you go about checking whether such a result is actually a root? One way would be to substitute that possible root for x in the original equation. Is the result of substitution now zero? If so, then yes, that x is a root. If no, then it's not.
I would use synthetic division as the fastest way in which to determine whether such an x value is a root. Are you familiar with synth. div.?

Answer 5

Your polynomial, x^3 + 2x^2 - x - 2, is of the 3rd order, due to that x^3 term. Thus, this polynomial should have 3 roots.

Answer 6

@mathmale could you not just do long division to solve this to though?

Answer 7

First find one factor then yes you can then use long division to get a trinomial in x^2 which can then be factored to give the other 2 factors.

Answer 8

Of course you could use long division! But synth. div. is much faster.

Answer 9

Let's determine whether 2 is or is not a root of the given poly. Set up synth. div. as follows: