What is the factorization of...
p^3 - 2p^2 -9p + 18?
if you do a little rearranging, you see
p^3-9p - 2p^2+18
p(p^2-9) - 2(p^2-9)
(p-2)(p^2-9)
(p-2)(p-3)(p+3)
Oh. I understand. That makes a lot of sense. Thanks!!
To find the factorization of the polynomial p^3 - 2p^2 - 9p + 18, we can use a method called synthetic division.
Step 1: List all the possible rational roots of the polynomial. In this case, the leading coefficient is 1 and the constant term is 18, so the possible rational roots are the factors of 18 divided by the factors of 1. That gives us ±1, ±2, ±3, ±6, ±9, and ±18.
Step 2: Use synthetic division to test these possible roots. Let's start with 1. Set up the synthetic division like this:
1 | 1 -2 -9 18
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1 -1 -10 8
Step 3: Look at the bottom row of numbers in the synthetic division result. If the last number is 0, then the corresponding factor is a root of the polynomial. In this case, the last number is 8, which is not zero.
Step 4: Repeat steps 2 and 3 with the other possible rational roots until you find one that gives a synthetic division result with a last number of 0. In this case, after trying different possible roots, we find that -3 gives a result with a last number of 0:
-3 | 1 -2 -9 18
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1 1 -6 12
Step 5: Now that we have found a root, we can write the factorization of the polynomial. The polynomial p^3 - 2p^2 - 9p + 18 can be factored as (p + 3)(p^2 + p - 6).
So, the factorization of p^3 - 2p^2 - 9p + 18 is (p + 3)(p^2 + p - 6).