If α and β are the Zeros of the Polynomial f(x)=6x^2+x−2, Find the Value of (α/β+β/α)

If α and β are the Zeros of the Polynomial f(x)=6x²+x−2, Find the Value of (α/β+β/α)

Solution

To find the value of (α/β+β/α), we need to determine the values of α and β.

The zeros of the polynomial f(x) = 6x² + x - 2 are the values of x for which f(x) = 0.

So, let's solve the equation:

6x² + x - 2 = 0

To solve the quadratic equation, we can use the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

For the given equation, a = 6, b = 1, and c = -2.

Using the quadratic formula, we have:

x = (-1 ± √(1² - 4(6)(-2))) / (2(6)) = (-1 ± √(1 + 48)) / 12 = (-1 ± √49) / 12 = (-1 ± 7) / 12

Therefore, we have two solutions:

x₁ = (-1 + 7) / 12 = 6/12 = 1/2

x₂ = (-1 - 7) / 12 = -8/12 = -2/3

Thus, α = 1/2 and β = -2/3.

Now, we can calculate (α/β+β/α):

(α/β + β/α) = (1/2) / (-2/3) + (-2/3) / (1/2) = (1/2) * (-3/2) + (-2/3) * (2/1) = -3/4 + (-4/3) = -9/12 + (-16/12) = (-9 - 16) / 12 = -25/12

Therefore, the value of (α/β+β/α) is -25/12.

Answer

If α and β are the zeros of the polynomial f(x)=6x²+x−2, then the value of (α/β+β/α) is -25/12.

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