The polynomial of degree \( 5, P(x) \) has leading coefficient 1 , has roots of multiplicity 2 at \( x=2 \) and \( x=0 \) and a root of multiplicity 1 at \( x=-5 \) Find a possible formula for \( P(x) \). \( P(x)= \)

To find a polynomial \( P(x) \) of degree 5 with the given conditions, we can start by using the information about the roots and their multiplicities. 1. **Identify the roots and their multiplicities:** - The root \( x = 2 \) has a multiplicity of 2. - The root \( x = 0 \) has a multiplicity of 2. - The root \( x = -5 \) has a multiplicity of 1. 2. **Construct the polynomial:** The polynomial can be expressed in factored form based on its roots: \[ P(x) = (x - 2)^2 (x - 0)^2 (x + 5) \] 3. **Simplify the expression:** We can rewrite \( (x - 0)^2 \) as \( x^2 \): \[ P(x) = (x - 2)^2 x^2 (x + 5) \] 4. **Expand the polynomial:** First, expand \( (x - 2)^2 \): \[ (x - 2)^2 = x^2 - 4x + 4 \] Now, we can multiply this by \( x^2 \): \[ (x^2 - 4x + 4)x^2 = x^4 - 4x^3 + 4x^2 \] Next, we multiply this result by \( (x + 5) \): \[ P(x) = (x^4 - 4x^3 + 4x^2)(x + 5) \] 5. **Distribute \( (x + 5) \):** \[ P(x) = x^4(x + 5) - 4x^3(x + 5) + 4x^2(x + 5) \] Expanding each term: - \( x^4(x + 5) = x^5 + 5x^4 \) - \( -4x^3(x + 5) = -4x^4 - 20x^3 \) - \( 4x^2(x + 5) = 4x^3 + 20x^2 \) 6. **Combine all the terms:** \[ P(x) = x^5 + 5x^4 - 4x^4 - 20x^3 + 4x^3 + 20x^2 \] Combine like terms: \[ P(x) = x^5 + (5x^4 - 4x^4) + (-20x^3 + 4x^3) + 20x^2 \] \[ P(x) = x^5 + x^4 - 16x^3 + 20x^2 \] Thus, a possible formula for \( P(x) \) is: \[ P(x) = x^5 + x^4 - 16x^3 + 20x^2 \]