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 · 2018-12-12T14:01:12+01:00

Answer:

Option D is correct.

Polynomial for the given zeroes are; [tex]f(x) = x^3+3x^2-5x-15[/tex]

Step-by-step explanation:

Given the roots of the polynomial function as:

[tex]x = -\sqrt{5}[/tex] , [tex]\sqrt{5}[/tex] 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:

[tex]x -(-\sqrt{5}) = x+\sqrt{5}[/tex]

[tex](x-\sqrt{5)}[/tex] and

[tex](x-(-3)) =x+3[/tex]

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

[tex]f(x) = (x+\sqrt{5})(x-\sqrt{5})(x+3)[/tex]

Using [tex](x-a)(x+a) = x^2-a^2[/tex]

we have;

[tex]f(x) = (x^2-(\sqrt{5})^2)(x+3)[/tex]

or

[tex]f(x)=(x^2-5)(x+3)[/tex]

[tex]f(x) = x^2(x+3) - 5 (x+3)[/tex]

using distributive property: [tex]a\cdot(b+c) = a\cdot b + a\cdot c[/tex]

[tex]f(x) = x^3+3x^2 - 5x - 15[/tex]

Therefore, the following polynomial is, [tex]f(x) = x^3+3x^2-5x-15[/tex]