Problem 32 - Algebra 2

Question

ENGINEERING. For Exercises 32 and \(33,\) use the following information. When a certain type of plastic is cut into sections, the length of each section determines its strength. The function \(f(x)=x^{4}-14 x^{3}+69 x^{2}-140 x+100\) can describe the relative strength of a section of length \(x\) feet. Sections of plastic \(x\) feet long, where \(f(x)=0,\) are extremely weak. After testing the plastic, engineers discovered that sections 5 feet long were extremely weak. Show that \(x-5\) is a factor of the polynomial function.

Step-by-Step Solution

Answer

\(x - 5\) is a factor because \(f(5) = 0\).

Step 1: Understand Factor Theorem

The Factor Theorem states that a polynomial \(f(x)\) has a factor \(x - c\) if and only if \(f(c) = 0\). In this exercise, we are tasked with proving that \(x - 5\) is a factor of \(f(x)\), which means we need to show that \(f(5) = 0\).

Step 2: Substitute x with 5

We need to calculate \(f(5)\) using the given polynomial function. Substitute 5 into the function: \(f(5) = 5^{4} - 14 \times 5^{3} + 69 \times 5^{2} - 140 \times 5 + 100\).

Step 3: Calculate Each Term

Calculate each term of \(f(5)\):\- \(5^{4} = 625\)\- \(-14 \times 5^{3} = -14 \times 125 = -1750\)\- \(69 \times 5^{2} = 69 \times 25 = 1725\)\- \(-140 \times 5 = -700\)\- Constant term: \(+100\).

Step 4: Sum the Values

Add the results from Step 3: \\(625 - 1750 + 1725 - 700 + 100 = 0\).

Step 5: Conclude Using Factor Theorem

Since \(f(5) = 0\), according to the Factor Theorem, this confirms that \(x - 5\) is indeed a factor of the polynomial function \(f(x) = x^{4} - 14x^{3} + 69x^{2} - 140x + 100\).

Key Concepts

Understanding Polynomial Functions

Polynomial functions are expressions consisting of variables and coefficients, structured with terms involving non-negative integer exponents of the variables. These functions can be seen as a collection of monomials. The organization of the terms typically varies from highest to lowest degree.

For example, in the polynomial function \( f(x) = x^4 - 14x^3 + 69x^2 - 140x + 100 \), each term represents a part of the equation:

• \( x^4 \) is the highest degree term indicating it is a quartic polynomial.
• \( -14x^3 \) shows the third-degree component.
• Lower degree terms include \( 69x^2 \), \( -140x \), and the constant \( 100 \).

Understanding the degree of a polynomial is essential because it determines the function's behavior, such as the shape of its graph and the potential number of roots or solutions.

Algebra in Solving Polynomial Equations

Algebra is the branch of mathematics dealing with symbols and rules for manipulating those symbols to solve equations and understand relationships between quantities. When it comes to polynomial functions, algebra is essential for solving equations and analyzing their behavior.

When dealing with a polynomial function such as \( f(x) = x^4 - 14x^3 + 69x^2 - 140x + 100 \), algebra is used to determine factors or roots. One way is by using the Factor Theorem. The theorem helps to decide whether a given binomial, like \( x - 5 \), is a factor. To confirm, you would substitute the root, \( x = 5 \), into the function and see if the result is zero:

• If \( f(c) = 0 \) for any value \( c \), then \( x-c \) is a factor of the polynomial.
• This process involves substituting the value into the polynomial and using algebraic simplification to show the result.

Using algebra, one can also divide the polynomial by its factors to find other roots or simplify the polynomial further.

Roots of Polynomials

Roots of a polynomial are the values of \( x \) that make the polynomial equal to zero. Finding these roots is crucial because they indicate where the polynomial intersects the x-axis on a graph. For example, given the polynomial function \( f(x) = x^4 - 14x^3 + 69x^2 - 140x + 100 \), determining the roots tells us when the polynomial equals zero.

To find the roots, one can:

• Use the Factor Theorem: By checking which values satisfy \( f(c) = 0 \), we can determine that the polynomial can be factored, such as \( x-5 \) for this exercise.
• Once a factor is known, use polynomial division to find the remaining factors and roots.
• Graphically, roots are where the curve cuts the horizontal axis, so they are often considered solutions to the polynomial equation.

Understanding the concept of roots allows for deeper insights into the nature of polynomial equations, predicting their graphs, and solving real-world problems related to them.