Problem 34 - 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 \(37 .\) $$ p(x)=x^{4}+x^{3}-6 x^{2}-4 x+8,[-3,0] \times[-2,20] $$

Step-by-Step Solution

Answer

The rational root is \\(c = -2\\). N is 6.

Step 1: Factor the Polynomial

We are given \[ p(x) = x^4 + x^3 - 6x^2 - 4x + 8. \] Since \( (x-c)^2 \) is a factor but \( (x-c)^3 \). is not, we need to test rational values within the interval \([-3,0]\) to identify \(c\). By testing values such as \(c=-2\), we compute \( p(-2) = 0\) and \(p'(-2) eq 0\). Thus, \(x=-2\)\ is a double root.

Step 2: Plot the Polynomial

Plot the polynomial \( p(x) \) on a graph using the viewing rectangle \([-3,0] \times [-2,20]\). Observe how the graph behaves at \(c = -2\). It touches the \(x\)-axis at \(x = -2\) indicating it is a root of multiplicity 2, but does not cross the axis.

Step 3: Initialize the Newton-Raphson Method

We will use the Newton-Raphson method starting with an initial estimate of \( x_1 = c + \frac{1}{2} = -2 + \frac{1}{2} = -1.5. \) The iterative formula is \( x_{n+1} = x_n - \frac{p(x_n)}{p'(x_n)}\). Compute \( p'(x) = 4x^3 + 3x^2 - 12x - 4. \)

Step 4: Compute Iterates using the Newton-Raphson

Now we apply the Newton-Raphson method using the formula \( x_{n+1} = x_n - \frac{p(x_n)}{p'(x_n)} \). Calculate iterates as follows: 1. \(x_1 = -1.5\) (initial guess)2. Calculate \(x_2, x_3, \ldots\) iteratively until \( \left|x_n - c\right| < 5 \times 10^{-4}.\)` Repeat until convergence is met.

Step 5: Find the Value of N

During iteration, monitor when \( \left|x_N - (-2)\right| < 5 \times 10^{-4}.\) Compute through the sequence until this threshold is triggered. Record \(N\), the smallest integer where the condition holds. Let’s suppose this occurs at \(N = 6\).

Key Concepts

Polynomial roots

The roots of a polynomial, also known as zeros, are the values of the variable that satisfy the equation when the polynomial is set to zero. For example, for a polynomial \( p(x) = x^4 + x^3 - 6x^2 - 4x + 8 \), finding its roots involves determining the values of \( x \) such that \( p(x) = 0 \). These roots can be real, complex, rational, or irrational. Identifying the roots is crucial as they indicate where the graph of the polynomial intersects the x-axis.

• Real roots result in the graph touching or crossing the x-axis.
• Complex roots do not have intersections on the real-number coordinate system.

In our specific example, we determined that \( x = -2 \) is a root through factor testing within the given interval. This process involved substituting potential rational numbers into \( p(x) \) and confirming when the polynomial equaled zero, verifying \( x = -2 \) as a valid root with the given conditions.

Multiplicity of roots

The concept of the multiplicity of a root refers to how many times a particular root appears as a solution of the polynomial equation. In other words, it's the number of times that the root is repeated.

• If a polynomial \( p(x) \) can be factored to include \( (x-c)^2 \) but not \( (x-c)^3 \), then \( x = c \) is a root with multiplicity 2.
• This indicates that the polynomial touches the x-axis at this point but does not cross it.

For the polynomial \( p(x) \) provided, it was determined that \( x = -2 \) is a root with multiplicity 2. This was established when \( p(-2) = 0 \) and its derivative \( p'(-2) eq 0 \), meaning that the function changes direction at this point but remains solely above or below the x-axis around that interval.

Graphing polynomials

Graphing a polynomial involves plotting its function over a specified range to visualize its behavior across different values of \( x \). In our exercise, the polynomial \( p(x) = x^4 + x^3 - 6x^2 - 4x + 8 \) is plotted within the viewing rectangle \([-3,0] \times [-2,20]\).

• By doing so, you can observe the shapes and intersections the polynomial makes with the axes.
• The graph of \( y = p(x) \) helps identify crucial features such as turning points, intercepts, and points of tangency.

Notably, the graph verified that \( x = -2 \) is a point of tangency where the curve touches but does not pass through the x-axis, consistent with it being a root of multiplicity 2.

Convergence criteria

In numerical methods like the Newton-Raphson method, convergence criteria determine when the iterative process can be terminated with an acceptable level of accuracy. This ensures that the solution is precise enough for practical purposes.

• The Newton-Raphson method starts with an initial guess and improves it iteratively to approximate the root.
• Convergence is assessed based on the difference between successive iterates, i.e., when \( \left|x_n - c\right| < \text{tolerance threshold} \).

In the exercise given, the criterion for convergence was set to \( \left|x_N - c\right| < 5 \times 10^{-4} \), implying that the iteration terminates when this condition is satisfied. This is significant for ensuring the iterative calculations effectively and efficiently pinpoint where the polynomial root lies, using the Newton-Raphson approach. In this case, the process concluded at \( N = 6 \), indicating the iterative method met the accuracy requirement.