Questions: State the number of complex zeros and the possible rational zeros for each function. Then find ALL zeros. - FOR #12, x=2 is a root. - FOR #13, x=-1 is a root. 11) f(x)=x^3+x^2-4x-4

State the number of complex zeros and the possible rational zeros for each function. Then find ALL zeros. - FOR #12, x=2 is a root. - FOR #13, x=-1 is a root. 11) f(x)=x^3+x^2-4x-4
Transcript text: State the number of complex zeros and the possible rational zeros for each function. Then find ALL zeros. - FOR \#12, $x=2$ is a root. - FOR \#13, $x=-1$ is a root. 11) $f(x)=x^{3}+x^{2}-4 x-4$
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Solution

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Solution Steps

Step 1: Factor the Polynomial

The polynomial \( f(x) = x^3 + x^2 - 4x - 4 \) can be factored as follows: \[ f(x) = (x - 2)(x + 1)(x + 2) \]

Step 2: Identify the Roots

From the factorization, we can identify the roots of the polynomial:

  • The root corresponding to \( (x - 2) \) is \( x = 2 \).
  • The root corresponding to \( (x + 1) \) is \( x = -1 \).
  • The root corresponding to \( (x + 2) \) is \( x = -2 \).
Step 3: List All Zeros

The complete set of zeros for the polynomial \( f(x) \) is: \[ \text{Zeros: } x = 2, \, x = -1, \, x = -2 \]

Step 4: Determine the Number of Complex and Rational Zeros

Since all the roots are real numbers, there are:

  • \( 0 \) complex zeros.
  • \( 3 \) rational zeros, which are \( 2, -1, \) and \( -2 \).

Final Answer

The number of complex zeros is \( \boxed{0} \) and the possible rational zeros are \( \boxed{2, -1, -2} \). All zeros are \( \boxed{2, -1, -2} \).

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