To find the real zeros of the polynomial $P(x) = x^3 + 3x^2 - 4$, we will use the Rational Root Theorem, which states that any rational root of a polynomial with integer coefficients must be a factor of the constant term divided by a factor of the leading coefficient.

1. Identify the coefficients:

   • The constant term ($a_0$) is $-4$.
   • The leading coefficient ($a_n$) is $1$.

2. List the possible rational roots:
   The factors of the constant term $-4$ are $\pm 1, \pm 2, \pm 4$. Since the leading coefficient is $1$, the possible rational roots are also $\pm 1, \pm 2, \pm 4$.

3. Test each possible root:
   We substitute each candidate into the polynomial and see if it results in zero.

   • $P(1) = 1^3 + 3(1)^2 - 4 = 1 + 3 - 4 = 0$
     So, $x = 1$ is a root.

4. Factor out $(x-1)$ from $P(x)$:
   Perform polynomial division of $P(x)$ by $(x-1)$.

   Dividing $x^3 + 3x^2 - 4$ by $(x-1)$, we get: $P(x) = (x-1)(x^2 + 4x + 4)$

5. Factor the quadratic $x^2 + 4x + 4$:
   The quadratic can be factored as: $x^2 + 4x + 4 = (x+2)(x+2) = (x+2)^2$

6. Write the fully factored form of $P(x)$:
   $P(x) = (x-1)(x+2)^2$

7. Identify the zeros:
   The zeros are $x = 1$ and $x = -2$. Since $(x+2)^2$ has a multiplicity of 2, $x = -2$ is a repeated root.

Thus, the zeros of the polynomial are: $1, -2, -2$

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Related Questions:

1. How do you perform polynomial division?
2. What is the Rational Root Theorem and how is it applied?
3. How do you find the multiplicity of a root in a polynomial?
4. How can you factor a cubic polynomial if you know one of its roots?
5. How do you solve a quadratic equation using factoring?

Tip:

When a root has a multiplicity greater than 1, the graph of the polynomial will touch the x-axis at that root without crossing it.