91Ó°ÊÓ

Factoring polynomials is a fundamental skill in algebra, especially when solving polynomial inequalities. It involves breaking down a polynomial into simpler components, or factors, that multiply together to give the original polynomial. This is particularly useful for identifying points where the polynomial might change behavior, such as intercepts on a graph.
In our exercise, the polynomial given was of the fourth degree: \(f(x) = x^4 + 3x^3 - 4x^2\). The first step was to identify common factors among the terms. We noticed each term contains \(x^2\) as a common factor, allowing us to factor it out and rewrite the function as \(f(x) = x^2(x^2 + 3x - 4)\).
After that, we focused on factoring the quadratic expression \(x^2 + 3x - 4\). The task is to find two numbers that multiply to the constant term (-4) and sum to the linear coefficient (3). After trying combinations, we find \(4\) and \(-1\) work, giving us the factorization: \(x^2(x + 4)(x - 1)\). This new form is crucial for further analysis, such as determining critical points.

Critical points occur where the polynomial equals zero, indicating potential sign changes. These points are crucial for understanding when the polynomial switches from positive to negative values, or vice versa. To find them, we set each factor from our factored polynomial equal to zero.
In our example, the factors \(x^2\), \(x + 4\), and \(x - 1\) provide the solutions for critical points:

• \(x^2 = 0\) leads to \(x = 0\).
• \(x + 4 = 0\) leads to \(x = -4\).
• \(x - 1 = 0\) leads to \(x = 1\).

So, the critical points are \(x = 0, -4,\) and \(1\). These points mark where the function \(f(x)\) touches or crosses the x-axis.
These points are not just random; they are essential in determining the intervals where the function is either positive (above the x-axis) or negative (below the x-axis).

Sign analysis helps us discern the nature of \(f(x)\) in intervals between the critical points. By choosing a test point in each interval, we determine whether \(f(x)\) is positive or negative.
Let's perform a sign analysis for our function, given intervals defined by the critical points \(x = -4, 0,\) and \(1\):

• For \((-\infty, -4)\), if we substitute a point like \(x = -5\), all factor signs are positive, making \(f(x) > 0\).
• For \((-4, 0)\), substituting \(x = -1\) results in \(x^2 > 0\), \(x + 4 > 0\), \(x - 1 < 0\), so \(f(x) < 0\).
• For \((0, 1)\), substituting \(x = 0.5\) gives \(x^2 > 0\), \(x + 4 > 0\), \(x - 1 < 0\), making \(f(x) < 0\).
• For \((1, \infty)\), by selecting \(x = 2\), all factors are positive, resulting in \(f(x) > 0\).

This methodical testing identifies where the function is positive, in \((-\infty, -4)\) and \((1, \infty)\), and negative, in \((-4, 0)\) and \((0, 1)\). Sign analysis is invaluable for sketching graphs and solving inequalities.

Graph sketching is a powerful tool for visualizing polynomial behavior. It provides a clear picture of how a polynomial behaves across different intervals. To sketch the graph of \(f(x) = x^2(x + 4)(x - 1)\), we need to combine our critical points and sign analysis.
First, plot the critical points \(-4, 0,\) and \(1\) on the x-axis, marking where the graph crosses or touches the axis. Then follow the behavior determined by sign analysis:
- From \(-\infty\) to \(-4\), the graph remains above the x-axis.
- Between \(-4\) and \(0\), it dips below the x-axis.
- Between \(0\) and \(1\), it remains below.
- Finally, after \(x = 1\), the graph goes above the x-axis.
Connecting these regions yields the complete graph, zigzagging above and below the x-axis. Graph sketching helps decode the function's true nature, revealing positive-negative intervals and where it glides across the x-axis.