How do you factor this?

Question

I'm stuck on this problem... 

x^4 - 11x^3 + 43x^2 - 65x 

Answer 1

The polynomial x⁴  - 11x³  + 43x² + 65x

= x(x³  - 11x²  + 43x + 65)

The function f(x) = x(x³  - 11x²  + 43x + 65)

0 is one of the zeros of above polynomial.

Now the find the zeros of  f(x) = x³  - 11x²  + 43x + 65 by using Rational Zero Theorem.

If p/q is a rational zero, then p is a factor of 65 and q is a factor of 1.

The possible values of p are   ± 1, ± 5, ± 13 and   ± 65.

The possible values for q are ± 1.

So, p/q = ± 1, ± 5, ± 13 and   ± 65.

Make a table for the synthetic division and test possible  zeros.

p/q	1	-11	43	65
-1	1	-12	55	-120
1	1	-10	33	-32
5	1	-6	13	0

Since f(5) = 0,  x = 5 is a zero. The depressed polynomial is x² - 6x + 13.

Comments

Contd..

Use the Quadratic Formula to find the roots of the related quadratic equation x² - 6x + 13 = 0.

Roots are

x = 3 + 2i and x = 3 - 2i

Zeros of x⁴  - 11x³  + 43x² + 65x are 0, 5, 3 + 2i and 3 - 2i

From the Factor theorem,

When f(c) = 0 then x - c is a factor of the polynomial.

Factoring of x⁴  - 11x³  + 43x² + 65x = x(x - 5)[x - (3 + 2i) ][x - (3 - 2i)].