Question

Find all the zeros of the polynomial x cube + 3 x square - 5 x minus 15 if two of its zeros are root 5 and minus root 5

Answer 1

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Answer 2

-3 is a root of the given polynomial.

Step-by-step explanation:

We are given the following in the question:

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

$x = \sqrt{5}, x = -\sqrt{5}$ are two roots of the given polynomial.

Since the polynomial have a degree three, there would be three zeros of the given polynomial.

We have to find the third zero of the polynomial.

$(x-\sqrt{5}), (x + \sqrt5)$ would be a factor of the given polynomial.

Thus, we can write:

$f(x) = x^3 + 3x^2 -5x - 15 = 0\\\text{To find the third root}\\\\g(x) = \dfrac{x^3 + 3x^2 -5x - 15}{(x+\sqrt5)(x-\sqrt5)}\\\\g(x) = \dfrac{x^3 + 3x^2 -5x - 15}{(x^2-5)}\\g(x) = x+3\\g(x) = 0\\\Rightarrow x = -3\\f(-3) = (-3)^3 + 3(-3)^2 -5(-3) - 15 = 0$

Thus, -3 is a root of the given polynomial.

#LearnMore

Find all the zeros of the polynomial 2 x power 4 - 10 x cube + 5 x square + 15 x minus 12 if it is given that two of its zeros are root 3 by 2 and minus root 3 by 2

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