Find the Zeros of the Polynomial x^2−3x−m(m+3)

Find the Zeros of the Polynomial x²−3x−m(m+3)

To find the zeros of the polynomial f(x) = x² – 3x – m(m + 3), we need to solve the equation f(x) = 0.

Let’s set up the equation:

x² – 3x – m(m + 3) = 0

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

x = (-b ± √(b² – 4ac)) / (2a)

For the given equation, a = 1, b = -3, and c = -m(m + 3).

Using the quadratic formula, we have:

x = (3 ± √((-3)² – 4(1)(-m(m + 3)))) / (2(1)) = (3 ± √(9 + 4m(m + 3))) / 2 = (3 ± √(9 + 4m² + 12m)) / 2 = (3 ± √(4m² + 12m + 9)) / 2 = (3 ± √((2m + 3)²)) / 2 = (3 ± (2m + 3)) / 2

This gives us two solutions:

x₁ = (3 + (2m + 3)) / 2 = (2m + 6) / 2 = m + 3 x₂ = (3 – (2m + 3)) / 2 = (-2m) / 2 = -m

Therefore, the zeros of the polynomial x² – 3x – m(m + 3) are x = m + 3 and x = -m.

Answer

Zeros of the Polynomial x²−3x−m(m+3) are x = m + 3 and x = -m

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