Question

Which of the following is a polynomial with roots - square root of 5, - square root of five and 3

A. X^3 - 3x^2 - 5x +15
B. X^3 + 2x^2 -3x - 6
C. X^3 - 2x^2 - 3x +6
D. X^3 + 3x^2 - 5x - 15

Answer 1

A is correct

Explanation:

What we need to do here is to multiply all the roots together

The roots are;

3, √5 and -√5

Let’s have them in form of a sum

if x = 3, then the root is x-3

If x = √5, then the root is x-√5

If x = -√5, then the root is x+ √5

Now we need to multiply all these together to arrive at the original polynomial

Let’s start by using the roots

(x-√5)(x+ √5)

we can use the difference of 2 squares here and we arrive at (x^2 -5)

So finally, the polynomial would be;

(x^2-5)(x-3)

= x(x^2-5) -3(x^2-5)

= x^3-5x-3x^2+15

By rearranging, we have;

x^3-3x^2-5x+15