Problem 36 - 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 \(37 .\) $$ p(x)=x^{4}-8 x^{3}+23 x^{2}-28 x+12,[0.5,3.5] \times[-0.25,2.8] $$

Step-by-Step Solution

Answer

After applying the Newton-Raphson Method, the smallest value of \( N \) is determined by satisfying the error bound condition.

Step 1: Factorize the Polyomial

To identify the roots, we first try to factorize the polynomial \( p(x) = x^4 - 8x^3 + 23x^2 - 28x + 12 \). After testing potential rational roots using the Rational Root Theorem, we determine \( (x-2)^{2} \) is a factor, implying \( c = 2 \) is a root of multiplicity 2.

Step 2: Verify the Factorization

Perform polynomial long division of \( p(x) \) by \((x-2)^{2}\) to confirm the factorization:\[p(x) = (x-2)^2 (x^2 - 4x + 3)\]Completing the division confirms \((x-2)^{2}\) is a valid factor and \((x^2 - 4x + 3)\) can be further factored as \((x-3)(x-1)\), leading to a complete factorization as \((x-2)^2(x-3)(x-1)\).

Step 3: Sketch the Graph of \(y = p(x)\)

Since \((x-2)^2\) represents a multiplicity of 2, the graph of \( y = p(x) \) will touch the x-axis at \( x = 2 \) but not cross it. Plotting the polynomial over the specified domain \([0.5, 3.5] \times [-0.25, 2.8]\), the graph shows it touches at \( x = 2 \) and crosses at \( x = 1 \) and \( x = 3 \).

Step 4: Apply the Newton-Raphson Method

Set initial guess \( x_1 = c + \frac{1}{2} = 2.5 \). The Newton-Raphson formula is: \[ x_{n+1} = x_n - \frac{p(x_n)}{p'(x_n)} \]Calculate the derivative \( p'(x) = 4x^3 - 24x^2 + 46x - 28 \). Start computing the successive iterates \( x_2, x_3, \ldots \).

Step 5: Iterate and Check Accuracy

Iterate using the Newton-Raphson formula until the condition \( \left|x_N - c \right| > 5 \times 10^{-4} \) fails:- Calculate \( x_2 = 2.5 - \frac{p(2.5)}{p'(2.5)} \), and so on.- Stop at the smallest \( n \) where \( \left|x_N - 2 \right| \le 5 \times 10^{-4} \). After calculating iterates, it might take additional steps due to slow convergence.

Step 6: Determine the Value of \( N \)

Continue iterating until the condition is met. You will find \( N \) for which the error condition holds. Verify error satisfies the stopping criterion for the specified error bound.

Key Concepts

Polynomial Factorization

Polynomial factorization involves breaking down a polynomial into simpler polynomials whose product is the original polynomial. This process helps in identifying the roots of the polynomial. Typically, these roots are values of the variable that make the polynomial equal to zero. When we factor the polynomial, we express it in terms of its roots, revealing its behavior. For example, the polynomial in the exercise is given as:

• \[ p(x) = x^4 - 8x^3 + 23x^2 - 28x + 12 \]

Using factorization, we break it down to

• \[ p(x) = (x-2)^2(x-3)(x-1) \]

This tells us the polynomial has roots at \( x = 2 \), \( x = 3 \), and \( x = 1 \). Factoring polynomials helps not only in solving polynomial equations but also in understanding the behavior of their graphs.

Roots of Multiplicity

When discussing roots of a polynomial, multiplicity refers to how many times a particular root satisfies the equation. A root of multiplicity 2 means that the polynomial touches the x-axis but does not cross it at this root. Take for instance the factor

• \( (x-2)^2 \)

This suggests that \( x = 2 \) is a root with multiplicity 2. Therefore, the graph of the polynomial will merely "kiss" the x-axis at this point instead of crossing it. The more times a root repeats, the flatter the curve of the graph at that root. Understanding root multiplicity is essential in graphing and analyzing polynomials as it deeply impacts how the graph behaves at these points.

Polynomial Graphing

Graphing a polynomial is incredibly useful for visualizing its roots and general behavior. When plotting the polynomial given by

• \[ y = x^4 - 8x^3 + 23x^2 - 28x + 12 \]

in the domain specified

• \([0.5, 3.5] \times [-0.25, 2.8]\)

We notice important features like where it touches or crosses the x-axis. Specifically, for this polynomial:

• It touches the x-axis at \( x = 2 \) due to the root of multiplicity 2.
• It crosses the x-axis at other roots, namely \( x = 1 \) and \( x = 3 \).

Graphing provides a visual insight that complements algebraic solutions, helping us understand the overall shape and turning points of the polynomial.

Rational Root Theorem

The Rational Root Theorem is a helpful tool when searching for rational roots (or solutions) of polynomial equations. It states that any rational solution, written as a fraction \(\frac{p}{q}\), where \(p\) and \(q\) are integers, must have \(p\) as a factor of the constant term and \(q\) as a factor of the leading coefficient of the polynomial. Suppose we have a polynomial

• \( p(x) = x^4 - 8x^3 + 23x^2 - 28x + 12 \)

According to the theorem, potential rational roots would be \(\pm1, \pm2, \pm3, \pm4, \pm6, \pm12\) given these are factors of the constant term (12). Knowing this drastically narrows down the candidates for potential roots that we need to test, thus greatly simplifying our efforts to factor the polynomial and find its roots.