For a given polynomial, a given zero is 2 + 5√3, what would be another zero? Ive tried to solve it myself but can't think of any ways to do it. If anyone could help, I'd appreciate it :)

2 - 5√3

Step-by-step explanation:

This is because for a polynomial aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ where n ≥ 2, there would always be a polynomial of at least degree n = 2.

Now, for a polynomial of degree n = 2, a₂x² + a₁x + a₀, the roots of the equation are in conjugate pairs if the discriminant a₁² - 4a₂a₀ is not a perfect square. So, we would have a surd in the root and thus a conjugate pair.

So given a zero of 2 + 5√3, its conjugate is 2 - 5√3 which is the other zero.

{x = -1/2 ,   y = -1

step-by-step explanation:

solve the following system:

{2 x + y = -2

{10 x - 3 y = -2

in the first equation, look to solve for y:

{2 x + y = -2

{10 x - 3 y = -2

subtract 2 x from both sides:

{y = -2 x - 2

{10 x - 3 y = -2

substitute y = -2 x - 2 into the second equation:

{y = -2 x - 2

{10 x - 3 (-2 x - 2) = -2

10 x - 3 (-2 x - 2) = (6 x + 6) + 10 x = 16 x + 6:

{y = -2 x - 2

{16 x + 6 = -2

in the second equation, look to solve for x:

{y = -2 x - 2

{16 x + 6 = -2

subtract 6 from both sides:

{y = -2 x - 2

{16 x = -8

divide both sides by 16:

{y = -2 x - 2

{x = -1/2

substitute x = -1/2 into the first equation:

{y = -1

{x = -1/2

collect results in alphabetical order:

answer:   {x = -1/2 ,   y = -1

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