Problem 66 How does the linear factorizatio... [FREE SOLUTION]

Chapter 3: Problem 66

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

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The linear factorization of a polynomial demonstrates that a polynomial of degree $n$ has $n$ roots because each term $(x-c_i)$ in the factorization corresponds to a root of the polynomial. Combined with the Fundamental Theorem of Algebra that guarantees $n$ roots for a degree-$n$ polynomial, it shows that every non-constant polynomial equation of degree $n$ has $n$ roots.

Step by step solution

HOV 鈥� The Fundamental Theorem of Algebra

The first thing to be aware of is the Fundamental Theorem of Algebra. This theorem states that every non-constant polynomial equation of degree $n$ has at least one root in the complex number plane. So, if our polynomial is of degree $n$, it is guaranteed to have $n$ roots, some of which may repeat and some may be complex.

What is Linear Factorization

Linear factorization is the process of breaking down a polynomial into a product of factors, where each factor is associated with a root of the polynomial. According to the notation used in this exercise, each factor of $f(x)$ is written as $(x-c_i)$, where $c_i$ is the $i$-th root of the polynomial.

Count the Roots

Observing the linear factorization of $f(x)$, each term $(x-c_i)$ corresponds to a different root of the polynomial. As such, it's clear there are $n$ separate terms. Therefore, given the Fundamental Theorem of Algebra, it can be concluded that an $n$-th degree polynomial will indeed have $n$ roots, whether real or complex, and these roots may repeat.

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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

Step by step solution

HOV 鈥� The Fundamental Theorem of Algebra

What is Linear Factorization

Count the Roots

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