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?

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.

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OrethaWilkison OrethaWilkison · Answer 2 · 2018-12-12T08:01:12-05:00

Answer:

Option D is correct.

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

Step-by-step explanation:

Given the roots of the polynomial function as:

\(x = -\sqrt{5}\) , \(\sqrt{5}\) 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:

\(x -(-\sqrt{5}) = x+\sqrt{5}\)

\((x-\sqrt{5)}\) and

\((x-(-3)) =x+3\)

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

\(f(x) = (x+\sqrt{5})(x-\sqrt{5})(x+3)\)

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

we have;

\(f(x) = (x^2-(\sqrt{5})^2)(x+3)\)

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