Question

Which of the following is a polynomial with roots negative square root of 5, square root of 5, and 3? (1 point)

Answer 1

The polynomial with roots negative square root of 5, square root of 5, and 3 is Option (C) $x^{3}$ - 3$x^{2}$ -5x +15 = 0 .

In the above question, it is given that  a polynomial with roots negative square root of 5, square root of 5, and 3

=> α = $-\sqrt{5}$

β = $\sqrt{5}$

ω = 3

We need to find the polynomial which can be made using the given roots of the polynomial

We know that, a cubic polynomial can be formed with the help of given roots as following

$x^{3}$ - ( α+β+ω)$x^{2}$ + (αβ+βω + αω)x - (αβω) = 0

So, first we'll find,

Sum of roots = α+β+ω = $-\sqrt{5}$+ $\sqrt{5}$ + 3 = 3

Product of two roots at a time = αβ+βω + αω = $-\sqrt{5}$$\sqrt{5}$ + $-\sqrt{5}.$3 + 3.$-\sqrt{5}$ =

-5 + $3\sqrt{5}$ $-3\sqrt{5}$ =  -5

Product of roots = αβω = $-\sqrt{5} . \sqrt{5} . 3$ = -15

Now putting all the values in the polynomial equation, we get

$x^{3}$ - ( α+β+ω)$x^{2}$ + (αβ+βω + αω)x - (αβω) = 0

$x^{3}$ - ( 3)$x^{2}$ + (-5 )x - (-15) = 0

$x^{3}$ - 3$x^{2}$ + (-5 )x - (-15) = 0

$x^{3}$ - 3$x^{2}$ -5x +15 = 0

Hence, the polynomial with roots negative square root of 5, square root of 5, and 3 is $x^{3}$ - 3$x^{2}$ -5x +15 = 0 .

To learn more about, cubic polynomial, here

brainly.com/question/28081769

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