x4+15x3+70x2+120x+64=0

Question

Numerical linear Algebra question

Answer 1

x⁴+15x³+70x²+120x+64=0 has coefficients 1, 15, 70, 120, 64.

If we add together the coefficients of even powers of x we get 1+70+64=135.

And, if we add together the remaining coefficients we get 15+120=135.

This shows that x=-1 is a root, which we can divide by, using synthetic division, to reduce the degree of the polynomial:

-1 | 1 15  70 120   64

      1  -1 -14 -56 | -64

      1 14  56   64 |   0 = x³+14x²+56x+64.

Note that x³+a³ can be factorised: (x+a)(x²-ax+a²), so if a=4:

x³+14x²+56x+64=x³+64+14x(x+4)=(x+4)(x²-4x+16)+14x(x+4).

We now have a common factor x+4, so we can factorise:

(x+4)(x²-4x+16+14x)=(x+4)(x²+10x+16)=(x+4)(x+2)(x+8).

So the roots are x=-1, -2, -4, -8.