The intervals where f(x)>0 for x^4-2x^3-3x^2

To find the intervals where \( f(x) > 0 \) for the function \( f(x) = x^4 - 2x^3 - 3x^2 \), we need to determine the regions where the function is positive. Here are the steps:

1. **Factor the function:** First, factor \( f(x) \) completely.

\[
f(x) = x^4 - 2x^3 - 3x^2 = x^2(x^2 - 2x - 3)
\]

Now, factor the quadratic expression:

\[
x^2 - 2x - 3 = (x - 3)(x + 1)
\]

So, the fully factored form is:

\[
f(x) = x^2 (x - 3) (x + 1)
\]

2. **Find the critical points:** Set \( f(x) = 0 \) to find the critical points.

\[
x^2 (x - 3) (x + 1) = 0
\]

This gives:

\[
x = 0, \quad x = 3, \quad x = -1
\]

3. **Determine the sign of \( f(x) \) in each interval:** We will test the sign of \( f(x) \) in the intervals determined by the critical points: \( (-\infty, -1) \), \( (-1, 0) \), \( (0, 3) \), and \( (3, \infty) \).

- For \( x \in (-\infty, -1) \): Choose \( x = -2 \).

  \[
  f(-2) = (-2)^2 \times (-2 - 3) \times (-2 + 1) = 4 \times (-5) \times (-1) = 20 \quad (\text{positive})
  \]

- For \( x \in (-1, 0) \): Choose \( x = -0.5 \).

  \[
  f(-0.5) = (-0.5)^2 \times (-0.5 - 3) \times (-0.5 + 1) = 0.25 \times (-3.5) \times 0.5 = -0.4375 \quad (\text{negative})
  \]

- For \( x \in (0, 3) \): Choose \( x = 1 \).

  \[
  f(1) = 1^2 \times (1 - 3) \times (1 + 1) = 1 \times (-2) \times 2 = -4 \quad (\text{negative})
  \]

- For \( x \in (3, \infty) \): Choose \( x = 4 \).

  \[
  f(4) = 4^2 \times (4 - 3) \times (4 + 1) = 16 \times 1 \times 5 = 80 \quad (\text{positive})
  \]

4. **Identify the intervals where \( f(x) > 0 \):**

From the analysis:

- \( f(x) > 0 \) in the intervals \( (-\infty, -1) \) and \( (3, \infty) \).

Therefore, the intervals where \( f(x) > 0 \) are:

\[
(-\infty, -1) \cup (3, \infty)
\]

Would you like more details or have any questions?

Here are some related questions:
1. How do we find the intervals where \( f(x) < 0 \)?
2. What are the turning points of \( f(x) = x^4 - 2x^3 - 3x^2 \)?
3. How can we sketch the graph of \( f(x) = x^4 - 2x^3 - 3x^2 \)?
4. What is the behavior of \( f(x) \) as \( x \) approaches infinity?
5. How do we determine the local minima and maxima of \( f(x) \)?

**Tip:** To check where a polynomial is positive or negative, factor it completely and test the sign of each factor in the intervals determined by the roots!

The intervals where f(x) > 0 for x^4 - 2x^3 - 3x^2

Explore how to find the intervals where the polynomial f(x) = x^4 - 2x^3 - 3x^2 is positive. This solution includes factorization, critical point analysis, and sign determination in different intervals. Ideal for students in Grades 10-12 studying advanced algebra and polynomial functions.

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