Question

Find an equation with a solution of x = 2 of multiplicity 1 , and a solution of x = − 1 of multiplicity 2 .
Write your answer in standard form.

Answer 1

We want to find an equation for the given solutions and the multiplicity of each solution.

The equation is:

$0 = (x - 2)*(x + 1)^2$

First, assume that we have a given equation and we know that we have solutions {x₁, x₂, ..., xₙ}, each one with multiplicity {m₁, ..., mₙ}.

The equation, of a polynomial that meets these requirements, is given by:

$0 = A*(x - x_1)^{m_1}*...*(x - x_n)^{m_n}$

Where A is the leading coefficient and can be any real number.

Now that we know that, here we have the solutions:

• x = 2 with multiplicity 2
• x = -1 with multiplicity 2

We don't have information about the leading coefficient, so we assume it is equal to 1.

Then the equation is:

$0 = (x - 2)*(x + 1)^2$

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