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Chapter 3: Problem 53

How does the linear factorization of \(f(x),\) that is, $$f(x)=a_{n}\left(x-c_{1}\right)\left(x-c_{2}\right) \cdots\left(x-c_{n}\right)$$ show that a polynomial equation of degree \(n\) has \(n\) roots?

Short Answer

Expert verified
The linear factorization of a polynomial function demonstrates that a polynomial of degree \(n\) has \(n\) roots because every linear term in the factorization equates to a solution, or root, of the polynomial equation.

Step by step solution

01

Understanding Polynomial Linear Factorization

Any polynomial function of degree \(n\) can be factored into a product of \(n\) linear terms. This is called the linear factorization. Each linear term in the product corresponds to a root of the polynomial equation. Consequently, a polynomial of degree \(n\) has \(n\) linear terms, implying that it has \(n\) roots.
02

Looking at the Formula

Looking at the given form of a polynomial \(f(x)\), it's clear that it's factored into linear terms: \(f(x) = a_n \cdot (x-c_1) \cdot (x-c_2) \cdot ... \cdot (x-c_n)\). Here, \(c_1, c_2, ..., c_n\) are the roots of the polynomial equation.
03

Connecting Roots and Factors

Each root corresponds to one factor in the equation. For instance, when \(x = c_1\), \(f(x) = a_n \cdot (x-c_1) = 0\). This is because any number multiplied by zero equals zero. The same relation exists between \(x = c_2\), \(x = c_3\), ..., and \(x = c_n\), proving that each factor corresponds to a root and thus a polynomial of degree \(n\) has \(n\) roots.

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Most popular questions from this chapter

Use the four-step procedure for solving variation problems given on page 356 to solve. The volume of a gas varies directly as its temperature and inversely as its pressure. At a temperature of 100 Kelvin and a pressure of 15 kilograms per square meter, the gas occupies a volume of 20 cubic meters. Find the volume at a temperature of 150 Kelvin and a pressure of 30 kilograms per square meter.

Write the equation of a rational function \(f(x)=\frac{p(x)}{q(x)}\) having the indicated properties, in which the degrees of \(p\) and \(q\) are as small as possible. More than one correct function may be possible. Graph your function using a graphing utility to verify that it has the required properties. \(f\) has vertical asymptotes given by \(x=-2\) and \(x=2, a\) horizontal asymptote \(y=2, y\) -intercept at \(\frac{9}{2}, x\) -intercepts at \(-3\) and \(3,\) and \(y\) -axis symmetry.

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