Question

Write the simplest polynomial function with the given roots -3,-2,1 and 5

Answer 1

The simplest polynomial function with the given roots is

$x^4-x^3-19x^2-11x+30$.

Solution:

Given roots of the polynomial are:

–3, –2, 1 and 5

If a is the root of the polynomial then the factor is (x –a).

So that, the factors of the given roots are:

(x –(–3)) = x + 3

(x –(–2)) = x + 2

(x – 1) = x – 1

(x – 5) = x –5

Polynomial = Product of the factors

                    = (x + 3)(x + 2)(x – 1)(x – 5)

Multiply first two factors.

                     $=(x^2+2x+3x+6)(x-1)(x-5)$

                     $=(x^2+5x+6)(x-1)(x-5)$

Now, multiply the next two factors.

                     $=(x^2+5x+6)(x^2-5x-x+5)$

                     $=(x^2+5x+6)(x^2-6x+5)$

Now multiply these two factors.

                    $=x^2(x^2-6x+5)+5x(x^2-6x+5)+6(x^2-6x+5)$

                     $=x^4-6x^3+5x^2+5x^3-30x^2+25x+6x^2-36x+30$

                     $=x^4-x^3-19x^2-11x+30$

The simplest polynomial function with the given roots is

$x^4-x^3-19x^2-11x+30$.