What are the zeros of the function f(x) = x^4 + 4x^3 + 3x^2, and what are their multiplicities?

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To find the zeros of the function f(x) = x^4 + 4x^3 + 3x^2, we first need to factor the expression.

First, observe that each term in the polynomial includes an x2 factor. We can factor this out:

f(x) = x^2(x^2 + 4x + 3)

Next, we can focus on the quadratic expression x^2 + 4x + 3. We need to find its roots using the factoring method or applying the quadratic formula. This quadratic can be factored as:

x^2 + 4x + 3 = (x + 1)(x + 3)

Now we have:

f(x) = x^2(x + 1)(x + 3)

To find the zeros, we solve:

  • x^2 = 0: The zero here is x = 0 with multiplicity 2, since it is squared.
  • x + 1 = 0: This gives us x = -1 with multiplicity 1.
  • x + 3 = 0: This results in x = -3 with multiplicity 1.

In summary, the zeros of the function and their multiplicities are:

  • Zero: 0 – Multiplicity: 2
  • Zero: -1 – Multiplicity: 1
  • Zero: -3 – Multiplicity: 1

Thus, the final result is:

  • Zer0s: 0, -1, -3
  • Multiplicities: 2, 1, 1

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