Problem 33 - Calculus Single Variable

Question

If \((x-c)^{2}\) is a factor of a polynomial \(p(x)\) but \((x-c)^{3}\) is not, then \(c\) is a root of \(p(x)\) of multiplicity \(2 .\) The graph of \(y=p(x)\) touches the \(x\) -axis at a root of multiplicity 2 but does not cross the \(x\) -axis there. Plot the given polynomial \(p(x)\) in the specified viewing rectangle. Identify a rational number \(c\) that is a root of \(p\) with multiplicity \(2 .\) Use the Newton-Raphson Method with initial estimate \(x_{1}=c+1 / 2\) to obtain iterates \(x_{2}, x_{3}, \ldots, x_{n} .\) Terminate the process at the smallest value of \(n\) for which \(\left|x_{N}-c\right|>5 \times 10^{-4}\). What is \(N ?\) You will notice that the convergence is slow. Record the value of \(N\) so that it can be used for comparison in Exercise \(37 .\) $$ p(x)=x^{4}-2 x^{3}+3 x^{2}-4 x+2,[-2,3] \times[-5,50] $$

Step-by-Step Solution

Answer

The rational root \(c\) is 1, with multiplicity 2. Using the Newton-Raphson method, the value of \(N\) is the number of iterations required until \(|x_N - 1|\) exceeds \(5 \times 10^{-4}\).

Step 1: Factor Identification

Recognize that \((x-c)^2\) is a factor of \(p(x)\), and \((x-c)^3\) is not, which indicates that \(c\) is a root of multiplicity 2.

Step 2: Root Identification via Graphing

Graph \(p(x) = x^4 - 2x^3 + 3x^2 - 4x + 2\) over the interval \([-2, 3] \times [-5, 50]\). Observe where the graph touches the x-axis without crossing it to identify potential rational roots of multiplicity 2.

Step 3: Graph Analysis

From graph inspection, note that the polynomial touches the x-axis at \(x = 1\), indicating \(c = 1\) is likely a root of multiplicity 2.

Step 4: Newton-Raphson Method Setup

Apply the Newton-Raphson method to approximate \(c\). Start with the initial guess \(x_1 = c + \frac{1}{2} = 1.5\), and use the formula: \(x_{n+1} = x_n - \frac{p(x_n)}{p'(x_n)}\).

Step 5: Derivative Calculation

First, find the derivative: \(p'(x) = 4x^3 - 6x^2 + 6x - 4\).

Step 6: Iterate Calculation

Calculate subsequent iterates: - Compute \(x_2 = 1.5 - \frac{p(1.5)}{p'(1.5)}\)- Continue iterating until the absolute difference \(|x_N - 1| < 5 \times 10^{-4}\).

Step 7: Iteration Result Analyzation

After several iterations, determine the smallest \(N\) such that the stopping condition is met.

Key Concepts

Polynomial Roots

When dealing with polynomials, roots are the solutions of the equation where the polynomial equals zero. These points on the graph of the polynomial exist along the x-axis where the graph touches or crosses. In the context of the exercise, if a polynomial is given as \(p(x)\), the equation \(p(x) = 0\) is essential for identifying roots. To better understand this concept, remember:

• Roots are the values of \(x\) for which \(p(x) = 0\).
• On the graph, these are x-values where the polynomial interacts with the x-axis.
• Finding roots can involve factoring, graphing, or computational methods like the Newton-Raphson method described in this exercise.

Finding polynomial roots is crucial for understanding the behavior of a polynomial, influencing both its shape and points of interaction with the x-axis.

Multiplicity of Roots

Multiplicity refers to the number of times a particular root appears as a factor in a polynomial. In simple terms, if a root \(c\) has multiplicity 2, then the factor \((x-c)^2\) is present in the polynomial. In our exercise:

• The polynomial \(p(x)\) has a root at \(x=1\) with multiplicity 2 as identified by \((x-1)^2\).
• Graphically, this means the curve touches the x-axis at \(x=1\) but does not cross it.
• When the multiplicity is greater than 1, the graph at the root is either flat or bounces off without crossing.

Understanding multiplicity helps in determining the shape of the graph at each root and how these points affect the overall graph.

Graphing Polynomials

Graphing polynomials provides a visual way to understand roots, multiplicities, and the overall behavior of the function. When considering a polynomial like \(p(x) = x^4 - 2x^3 + 3x^2 - 4x + 2\), plotting over a specified interval allows you to:

• Observe where the graph intersects or touches the x-axis, identifying roots.
• Understand the polynomial's turning points and symmetry.
• Visualize root multiplicities; roots with odd multiplicity cross the x-axis and even ones touch and turn.

For this exercise, we graph the polynomial in the interval \([-2, 3] \times [-5, 50]\), providing clues such as the root \(x=1\) where the polynomial touches but doesn't cross the axis, indicating its multiplicity.

Derivative Calculation

Calculating the derivative of a polynomial is a key part of solving and analyzing polynomials, especially when utilizing methods like Newton-Raphson for root approximation. The derivative, denoted \(p'(x)\), provides critical insights:

• Helps in determining the slope of the tangent at any point on the graph of the polynomial.
• Useful for Newton-Raphson calculations, where iteratively finding roots involves the equation \(x_{n+1} = x_n - \frac{p(x_n)}{p'(x_n)}\).
• In the exercise, finding \(p'(x) = 4x^3 - 6x^2 + 6x - 4\) assists in understanding both the shape of the graph and the steepness at different points.

Thus, derivative calculation is an essential mathematical tool in polynomial analysis, helping solve equations and analyze graph characteristics efficiently.