Question

Which of the following is a polynomial with roots - square root of 5, square root of 5, and -3 ? x3 - 2x2 - 3x + 6, x3 + 2x2 - 3x - 6, x3 - 3x2 - 5x + 15, or x3 + 3x2 - 5x - 15?

Answer 1

Option D is correct.

Polynomial for the given zeroes are;
`f(x) = x^3+3x^2-5x-15`

Explanation:

Given the roots of the polynomial function as:

`x = -√(5)` ,
`√(5)` and -3.

A root of a polynomial function is a number that, when plugged in for the variable, makes the function equal to 0.

First find the factors, we subtract the roots

so factors are:

`x -(-√(5)) = x+√(5)`

`(x-√(5))` and

`(x-(-3)) =x+3`

Now, to find the general polynomial f(x) we multiply these factors as:

`f(x) = (x+√(5))(x-√(5))(x+3)`

Using
`(x-a)(x+a) = x^2-a^2`

we have;

`f(x) = (x^2-(√(5))^2)(x+3)`

or

`f(x)=(x^2-5)(x+3)`

`f(x) = x^2(x+3) - 5 (x+3)`

using distributive property:
`a\cdot(b+c) = a\cdot b + a\cdot c`

`f(x) = x^3+3x^2 - 5x - 15`

Therefore, the following polynomial is,
`f(x) = x^3+3x^2-5x-15`

Answer 2

The fourth option.

The roots of x^3 + 3x^2 - 5x - 15 are - 3, +√5 and -√5.

To find the roots you can factor the polynomial in this way:

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

The roots are the values of x that make the function = 0.

Then the roots are

x + 3 = 0 ==> x = 3, and

x^2 - 5 = 0 ==> x = +/- √5.