Answer to Question #90887 in Algebra for Gaurav

Question

Find the roots of the equation
g(x) = x4+ 7x3+ 11x2+ 7x + 10 = 0.
Given that (x2+ 1) divides g(x).

Answer 1

Firstly we should write the integer divisions of the constant (free term). From them we can find out the first root of the equation.

D(10) = ±1, ±2, ±5, ±10.

We have found out, that the first root is -2. Now we should use the Horner`s method. We should make a table with 2 lines and 6 columns. In the first line we write down all coefficients of the equation from the second cell, in the second line we write down such numbers: in the first cell - the root, in the second - 1, in cells number n we write down the result of the operation (the root (-2) * number in the cell n-1 from the 2nd line + number in the cell n from the 1st line).

***1 7 11 7 10

-2 1 5 1 5 0

(I wrote *** to show how the table must look like)

The new equation is:

x³ + 5x² + x + 5 = 0;

We have found out the next root - -5. And we should use Horner`s method again:

***1 5 1 5

-5 1 0 1 0

The new equation is:

x² + 1 = 0;

x² = -1 - the equation doesn`t have the roots.

So the roots are:

x₁ = -2;

x₂ = -5.

Now we can decompose the polynomial and do the division:

g(x) = x⁴ + 7x³ + 11x² + 7x + 10 = (x²+1)(x+2)(x+5);

(x² + 1) / g(x) = (x²+1) / (x²+1)(x+2)(x+5) = 1 / (x+2)(x+5).