Question

solve x^4 +4x^3-36x^2+364x-957=0

I don't know how to do these kind of problems. Can someone help me?

Thx

Answer 1

First look for rational zeroes. 957=3×11×29 so the rational zeroes would be ±3, ±11, ±29.

So by plugging in each of these 6 possible values for x we can see if the equation is satisfied.

Start with x=3: 81+4×27-36×9+364×3-957=81+108-324+1092-957=0, so x=3 is a root.

Using synthetic division, divide by this root:

3 | 1 4  -36 364  -957

     1 3   21  -45 | 957

     1 7  -15 319 |  0 = x³+7x²-15x+319

319=11×29=319 so now try x=11:

1331+847-165+319=2332 and x=-11: -1331+847+165+319=0, so x=-11 is a root. Divide by this:

-11 | 1    7  -15    319

        1 -11   44 | -319

        1  -4    29 |     0 = x²-4x+29, which can be solved using the quadratic formula, or:

x²-4x=-29, x²-4x+4=-25, (x-2)²=-25, x-2=±5i, so the complex roots are x=2+5i and 2-5i.

COMPLETE SOLUTION

x=3, -11, 2+5i, or 2-5i.