Question

When a certain type of plastic is cut into sections, the length of each section determines its relative strength. The function \(f(x)=x^{4}-14 x^{3}+69 x^{2}-140 x+100\) describes the relative strength of a section of length \(x\) feet. After testing the plastic, engineers discovered that 5 -ft sections were extremely weak. a) Why is \(x-5\) a possible factor when \(x=5\) is the length of the pipe? Show that \(x-5\) is a factor of the polynomial function. b) Are there other lengths of plastic that are extremely weak? Explain your reasoning.

Short answer

a) \( x-5 \) is a factor because \( f(5)=0 \). b) Other weak lengths can be found by identifying other roots.

Step-by-step solution

Understanding the Problem

The problem gives a polynomial function describing the relative strength of a plastic section of length \( x \). The function is \( f(x)=x^{4}-14x^{3}+69x^{2}-140x+100 \). We need to show that \( x-5 \) is a factor when \( x = 5 \) and find other weak section lengths.

Factor Theorem Application

According to the Factor Theorem, if \( x-5 \) is a factor of \( f(x) \), then \( f(5) \) should equal zero. Calculate \( f(5) \):

Calculation

Substitute \( x = 5 \) into \( f(x) \): \[ f(5) = (5)^4 - 14(5)^3 + 69(5)^2 - 140(5) + 100 \] Simplify it step-by-step: \[ = 625 - 1750 + 1725 - 700 + 100 \] \[ = 625 - 1750 + 1725 - 700 + 100 = 0 \] This shows \( f(5) = 0 \), proving that \( x-5 \) is indeed a factor.

Check for Other Factors

To find other lengths that are extremely weak, consider other factors of the polynomial. This can be done through polynomial factorization or synthetic division. Look for values that make \( f(x) = 0 \). Since exact methods might vary, trial and testing known roots could be efficient.

Synthetic Division for Roots

Use synthetic division to test other potential roots (like 1, 2, etc.), or possibly use factorization techniques for polynomials. Each root found will represent a length \( x \) at which the section becomes weak (\( f(x) = 0 \)).

Key concepts

Factor Theorem

The Factor Theorem is a valuable tool in polynomial algebra. It states that if you have a polynomial function, say \(f(x)\), and a value \(c\), then \(x-c\) is a factor of \(f(x)\) if and only if \(f(c) = 0\). This theorem connects the concepts of zeros or roots of a polynomial with its factors.
For example, consider the polynomial given in the problem: \(f(x) = x^4 - 14x^3 + 69x^2 - 140x + 100\). If we know that a 5-ft section of plastic is weak, it's telling us that \(f(5) = 0\). Therefore, by the Factor Theorem, \(x-5\) must be a factor of \(f(x)\).
This principle helps simplify polynomials and find their roots or zeros, making it easier to solve complex polynomial equations.

Polynomial Roots

Root-finding is essential in understanding the behavior of polynomial functions. A root of a polynomial is a value \(x = c\) for which \(f(c) = 0\). These are points where the polynomial crosses or touches the x-axis. Roots are also known as zeros of the polynomial. In the given problem, we've already identified \(x = 5\) as a root because \(f(5) = 0\).
To find other roots of the polynomial \(f(x) = x^4 - 14x^3 + 69x^2 - 140x + 100\), methods such as factoring, the Rational Root Theorem, synthetic division, or numerical approaches like Newton's method can be used.
When engineers search for other weak lengths of plastic, they essentially look for other roots or zeros of the polynomial. These roots signify lengths \(x\) where the function \(f(x) = 0\), indicating a weak section.

Synthetic Division

Synthetic division simplifies the division of a polynomial by a binomial of the form \(x - c\). It is a streamlined alternative to long division in algebra. This technique also helps in evaluating polynomials and finding roots.
Here's a basic breakdown of how synthetic division works with our example polynomial \(f(x)=x^4 - 14x^3 + 69x^2 - 140x + 100\) and checking \(x = 5\):

• Write down the coefficients: \([1, -14, 69, -140, 100]\).
• Place the value \(5\) outside the division bracket.
• Bring down the leading coefficient (1) to the bottom row.

Synthetic division reveals whether \(x=5\) is a root. If the remainder is zero, then \(x-5\) is a factor.
By repeating this process with other potential roots, we can quickly test numerous possible values without performing tedious calculations.

Relative Strength Function

The relative strength function in this problem is \( f(x) = x^4 - 14x^3 + 69x^2 - 140x + 100 \). It models how the strength of a plastic section changes with its length. The polynomial function's values for different lengths (\(x\)) can be analyzed to determine points of weakness or strength.
When engineers say a 5-ft section is weak, they're indicating that \(f(5) = 0\), showing zero strength. By exploring this function further and finding additional roots, one can pinpoint other lengths (values of \(x\)) where the relative strength is zero, signifying weak sections. This understanding aids in improving the design and choosing appropriate materials for different lengths.
Manipulating such polynomials through factorization and synthetic division reveals critical insights into the material properties and helps in making informed engineering decisions.