Find the remaining factors of the polynomial 2x4−15x3+11x2+78x−40

Question

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Answer 1

To find the remaining factors of the polynomial 2x^4 - 15x^3 + 11x^2 + 78x - 40, we can use polynomial division or synthetic division to divide the polynomial by a known factor.

Let's start by trying to divide by (x - 2) since we can easily guess that x = 2 is a potential root. Using synthetic division, we get:

2 | 2 -15 11 78 -40
-4 2 26 208
---------------------
2 -19 13 104 168

The remainder after dividing by (x - 2) is 168. This means that (x - 2) is not a factor of the polynomial.

Now, let's try dividing the polynomial by another potential root, (x + 1):

-1 | 2 -19 13 104 168
-2 21 -34 -70
---------------------
2 -21 34 70 98

The remainder after dividing by (x + 1) is 98. This means that (x + 1) is not a factor of the polynomial.

Since we have exhausted the potential rational roots without finding any that work, the remaining factors of the polynomial are the quadratic factors that cannot be further factored using rational roots.

Therefore, the remaining factors of the polynomial 2x^4 - 15x^3 + 11x^2 + 78x - 40 are:

(x - 2), (x + 1), and the quadratic factors that cannot be determined with the given information.