Problem 3

Question

True or false? If $c$ is a real zero of the polynomial $P,$ then all the other zeros of $P$ are zeros of $P(x) /(x-c) .$

Step-by-Step Solution

Verified

Answer

True. All other zeros of $ P $ are found in $ P(x)/(x-c) $ as factors.

1Step 1: Understanding the Problem

We are given that if $ c $ is a real zero of a polynomial $ P(x) $, we need to determine whether all the other zeros of $ P $ are zeros of $ \frac{P(x)}{x-c} $. Let's evaluate this by understanding the relationship between polynomials and their factors.

2Step 2: Using Polynomial Division

If $ c $ is a zero of $ P(x) $, then $(x - c)$ is a factor of $ P(x) $. Hence, $ P(x) $ can be expressed as $ (x - c)Q(x) $ where $ Q(x) $ is another polynomial. Here, $ \frac{P(x)}{x-c} = Q(x) $.

3Step 3: Identifying Zeros of Q(x)

Since $ P(x) = (x - c)Q(x) $ incorporates all the zeros of $ P(x) $, the polynomial $ Q(x) $ contains all the zeros of $ P(x) $ except $ c $. Therefore, if $ \alpha $ is a zero of $ P(x) $ and $ \alpha eq c $, then $ \alpha $ must satisfy $ Q(\alpha) = 0 $.

4Step 4: Evaluating the Given Statement

According to our analysis, all zeros of $ P(x) $ except $ c $ must also be zeros of $ Q(x) $, which is $ \frac{P(x)}{x-c} $. Thus, it follows that the statement is true.

Key Concepts

Real ZerosPolynomial FactorsZeros of a Polynomial

Real Zeros

In polynomial mathematics, real zeros are the values of $ x $ that make the polynomial equal to zero. Finding a real zero means identifying a value $ c $ where $ P(c) = 0 $. These zeros are crucial since they represent the solutions to the polynomial equation. Real zeros are the points where the graph of the polynomial touches or crosses the x-axis, giving a clear visual representation of these important values.
If $ c $ is a real zero of a polynomial $ P(x) $, then $ (x-c) $ is a factor of the polynomial. Using polynomial division, you can factor the polynomial by dividing $ P(x) $ by $ (x-c) $. This division is fundamental in simplifying polynomials and finding additional zeros, especially when dealing with large and complex expressions.

Real zeros provide insight into the behavior of polynomials.
Their identification is essential for graphing and solving polynomial inequalities.
They serve as the starting point for deeper polynomial analysis, such as factorization.

Polynomial Factors

Polynomial factors are expressions that, when multiplied together, produce a given polynomial. For a polynomial $ P(x) $, if $ (x-c) $ is a factor, then $ c $ is a zero of the polynomial. The factorization of polynomials simplifies them, revealing their structure and making it easier to find their zeros.
Polynomial division is a technique used to divide a polynomial by one of its factors. When you divide $ P(x) $ by $ (x-c) $, and $ c $ is indeed a zero, the remainder is zero, and the division yields a polynomial called $ Q(x) $. This polynomial $ Q(x) $ containts the other factors of $ P(x) $.

Factoring breaks down polynomials into simpler components.
It helps find other zeros and the end behavior of the polynomial.
Useful in reducing polynomial equations to simpler forms.

Zeros of a Polynomial

Zeros of a polynomial are the values for which the polynomial equals zero. These zeros include both real and complex numbers, forming the complete set of solutions to the polynomial equation. Understanding zeros is integral to solving polynomial equations and analyzing polynomials.
Finding the zeros involves several methods, such as factoring, using the quadratic formula for second-degree polynomials, or even using synthetic division for more complex polynomials. The Fundamental Theorem of Algebra states that a polynomial of degree $ n $ has exactly $ n $ zeros, including complex zeros, counted with multiplicities.

Zeros are critical for determining polynomial roots.
They assist in graphically interpreting the polynomial's behavior.
Provide a complete picture of the polynomial's solution set.

In the discussed exercise, $ Q(x) $ represents the polynomial obtained by removing one zero, $ c $, and hence contains all other zeros of $ P(x) $. This approach helps in finding solutions more efficiently by reducing the polynomial to a simpler form.

Problem 2

Problem 3

Other exercises in this chapter

Problem 2

(a) If we divide the polynomial $P(x)$ by the factor $x-c$ and we obtain a remainder of 0 , then we know that $c$ is a ________of $P$

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Problem 2

Every polynomial has one of the following behaviors: $$\begin{array}{l}{\text { (i) } y \rightarrow \infty \text { as } x \rightarrow \infty \text { and } y \ri

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Problem 3

$3-8=$ Two polynomials $P$ and $D$ are given. Use either synthetic or long division to divide $P(x)$ by $D(x),$ and express $P$ in the form \(P(x)=D

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Problem 3

A polynomial of degree $n \geq 1$ has exactly ________ zeros if a zero of multiplicity m is counted m times.

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