Problem 20 - Algebra and Trigonometry

Question

Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior. $$ P(x)=(x-1)^{2}(x+2)^{3} $$

Step-by-Step Solution

Answer

The graph touches the x-axis at \(x = 1\), passes through \(x = -2\), with end behavior down as \(x\to-\infty\) and up as \(x\to+\infty\).

Step 1: Identify the Roots and Their Multiplicities

The polynomial \( P(x) = (x-1)^2(x+2)^3 \) has roots where each factor is set to zero. This gives us the roots: \( x = 1 \) and \( x = -2 \). The root \( x = 1 \) has a multiplicity of 2, meaning the graph will touch the x-axis and turn around at this point. The root \( x = -2 \) has a multiplicity of 3, indicating that the graph will pass through the x-axis, changing direction, and create an inflection point at this intercept.

Step 2: Determine the End Behavior

The end behavior of the polynomial is determined by the leading term when the expression is fully expanded. The highest power term will be \( x^5 \), since \((x-1)^2(x+2)^3\) expanded gives an overall degree of 5. For polynomials with odd degree and positive leading coefficient, like ours, the graph will fall to negative infinity as \( x \) approaches negative infinity, and it will rise to positive infinity as \( x \) approaches positive infinity.

Step 3: Calculate and Plot the Y-Intercept

To find the y-intercept, evaluate \( P(x) \) at \( x = 0 \). Substitute into the polynomial: \( P(0) = (0-1)^2(0+2)^3 = 1 \times 8 = 8 \). Thus, the y-intercept is (0, 8). This point is plotted on the graph.

Step 4: Sketch the Graph

With all this information, you can sketch the graph:1. Start at \( x = -\infty \), the graph goes to \(-\infty\), and approaches \( x = -2 \).2. At \( x = -2 \), the graph crosses the x-axis with an inflection point.3. As \( x \) moves towards 1, the graph comes up and touches the x-axis at \( x = 1 \) and turns around.4. As \( x \) continues to rise, the graph goes through the y-intercept at (0, 8) and then rises towards \( +\infty \) as \( x \to +\infty \).

Key Concepts

polynomial roots

Polynomial roots are the points where the polynomial evaluates to zero. These are crucial as they represent where the graph intersects the x-axis. For our polynomial \( P(x) = (x-1)^2(x+2)^3 \), the factors \((x-1)^2\) and \((x+2)^3\) indicate two roots: \( x = 1 \) and \( x = -2 \). Root \( x = 1 \) shows up when \((x-1) = 0\), and \( x = -2 \) appears when \((x+2) = 0\). Understanding these roots helps explain the overall shape and direction of the polynomial graph. When sketching, you'll plot these points as the locations where the graph interacts with the axis. These roots give the primary structure to the graph, showing where it touches or crosses the x-axis.

multiplicity of roots

The multiplicity of a root refers to how many times a particular root appears within the polynomial. This concept is essential because it influences the shape and behavior of the graph at the root.For \((x-1)^2\), the root \( x = 1 \) has a multiplicity of 2. This multiplicity tells us that at \( x = 1 \), the graph touches the x-axis and then turns back. It shows that the point can be a local minimum or maximum depending on the polynomial's coefficient.The other root, \( x = -2 \), with a factor \((x+2)^3\), has a multiplicity of 3. In this case, the graph will not just touch the x-axis but cross it, changing direction. This creates an inflection point at \( x = -2 \). The higher multiplicity means more pronounced behavior at this root. The polynomial curve slightly flattens as it approaches, crosses, and leaves the x-axis here.

end behavior of polynomials

End behavior describes how the polynomial graph behaves as \( x \) heads towards infinity, either positive or negative. This behavior is mostly defined by the polynomial's leading term. For \( P(x) = (x-1)^2(x+2)^3 \), expanding gives a dominant term of \( x^5 \). A degree 5 polynomial has an odd degree, and since it's positive, as \( x \to +fty \), \( P(x) \to +\infty \). Conversely, as \( x \to -\infty \), \( P(x) \to -\infty \). It indicates that:

• When \( x \) approaches negative infinity, the graph falls towards negative infinity.
• When \( x \) approaches positive infinity, the graph rises towards positive infinity.

Knowing the end behavior is essential for sketching the graph as it sets the boundaries and helps predict the general outline of the polynomial.

y-intercept calculation

The y-intercept is the point where the polynomial graph crosses the y-axis. It gives insight into the vertical position of the graph. To find it, substitute \( x = 0 \) into the polynomial equation. For \( P(x) = (x-1)^2(x+2)^3 \), calculate \( P(0) = (0-1)^2(0+2)^3 \). Solving this,

• \( (0-1)^2 = 1 \)
• \( (0+2)^3 = 8 \)
• Thus, \( P(0) = 1 \times 8 = 8 \)

This tells us that the y-intercept is at the point \( (0, 8) \). When sketching, you plot this point on the graph to help in accurately drawing the curve.