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Chapter 4: Problem 1

\(1-12\) . A polynomial \(P\) is given. (a) Find all zeros of \(P\) , real and complex. (b) Factor \(P\) completely. $$ P(x)=x^{4}+4 x^{2} $$

Short Answer

Expert verified
Zeros: 0, 2i, -2i. Factorization: \(x^2(x-2i)(x+2i)\).

Step by step solution

01

Recognize the structure of the polynomial

The given polynomial is \( P(x) = x^4 + 4x^2 \). Notice that it can be rewritten in a form that suggests factoring: \( P(x) = x^2(x^2 + 4) \). This hints at possible factoring by grouping or substitutions.
02

Factor using substitution

Let \( y = x^2 \). Then the polynomial becomes \( y(y + 4) = y^2 + 4y \). This is easier to factor: \( y(y + 4) = x^2(x^2 + 4) \). Thus, the polynomial is factored as \( P(x) = x^2(x^2 + 4) \).
03

Solve for zeros from the factored form

From the factored form \( x^2(x^2 + 4) = 0 \), set each factor equal to zero. Solve \( x^2 = 0 \) to get \( x = 0 \). Next, solve \( x^2 + 4 = 0 \), which gives \( x^2 = -4 \). Thus, \( x = \pm 2i \). These are the complex zeros of the polynomial.
04

Compile all the zeros

The zeros of the polynomial \( P(x) \) are \( x = 0, 2i, \text{ and } -2i \). There are no additional real zeros beyond \( x = 0 \).
05

Write the complete factorization

Using the zeros, we can write the fully factored polynomial as \( P(x) = x^2(x-2i)(x+2i) \). Since \( x^2 \) is already a factor, \( x \) as a zero is included here. The solutions for \( x^2 + 4 = 0 \) provide the complex factors.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Factorization
Polynomial factorization is the method of breaking down a given polynomial into simpler polynomials, called factors, that when multiplied together give the original polynomial. This is similar to breaking down numbers into their prime factors. It helps reveal the zeros of the polynomial and simplifies solving equations or performing integrations.
  • **Recognizing Factorable Terms**: The first step is to observe the given polynomial for opportunities to factor. In our equation, \( P(x) = x^4 + 4x^2 \), observe that it can be expressed as \( x^2(x^2 + 4) \).
  • **Using Substitution**: By letting \( y = x^2 \), our polynomial simplifies to \( y(y + 4) \), a particularly easy form to factor, leading us back to \( x^2(x^2 + 4) \).
Using polynomial factorization, we can simplify complex algebraic computations and identify important characteristics like zeros. This is fundamental in deeper mathematical analysis.
Complex Numbers
Complex numbers extend the idea of the one-dimensional number line (real numbers) to the two-dimensional plane by introducing an imaginary unit, usually denoted as \( i \), where \( i^2 = -1 \). This allows us to find solutions for equations that have no real numbers solutions.
  • **Understanding Complex Roots**: In the polynomial \( x^2 + 4 = 0 \), solving for \( x \) yields \( x = \pm 2i \). This is because the square of a real number cannot be negative, so we employ complex numbers instead.
  • **Significance in Polynomials**: Finding complex roots is crucial because it provides a complete set of solutions for polynomials, which often involves imaginary numbers alongside real numbers.
Complex numbers thus enable us to handle a broader spectrum of mathematical challenges, including complete polynomial solutions and transformations.
Factoring Techniques
Factoring techniques in algebra are methods used to simplify polynomials by expressing them as a product of simpler polynomials. These techniques are powerful tools in unraveling complex algebraic expressions and include several strategies.
  • **Factoring by Grouping**: This method involves grouping terms with common factors and simplifying. In our polynomial, \( x^4 + 4x^2 \), it was rewritten as \( x^2(x^2 + 4) \) to allow for simpler factorization.
  • **Using Substitution**: Often applied when a polynomial can be simplified by replacing variables, making the expression easier to factor, as seen with substituting \( y = x^2 \).
  • **Recognizing Patterns**: Identifying and leveraging common algebraic identities such as difference of squares, perfect square trinomials, and sum/difference of cubes can help efficiently break down expressions.
These techniques streamline problem-solving by reducing the complexity of polynomial expressions.
Real and Complex Zeros
The zeros of a polynomial function are the solutions to the equation \( P(x) = 0 \). These zeros can be real or complex numbers, and identifying them is crucial in understanding the polynomial's behavior:
  • **Real Zeros**: These are x-values where the polynomial crosses or touches the x-axis. In \( P(x) = x^4 + 4x^2 \), \( x = 0 \) is a real zero where the graph touches the x-axis.
  • **Complex Zeros**: When real solutions don't exist, we find complex zeros which occur in conjugate pairs. Here, from \( x^2 + 4 = 0 \), the complex zeros are \( x = 2i \) and \( x = -2i \).
Understanding the nature of these zeros allows us to factor the polynomial completely and understand its graph and properties. Whether real or complex, zeros provide essential insights into polynomial functions.

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Most popular questions from this chapter

\(41-58=\) Find all zeros of the polynomial. $$ P(x)=4 x^{4}+2 x^{3}-2 x^{2}-3 x-1 $$

In Example 2 we saw that some simple rational functions can be graphed by shifting, stretching, or reflecting the graph of \(y=1 / x .\) In this exercise we consider rational functions that can be graphed by transforming the graph of \(y=1 / x^{2},\) shown on the following page. (a) Graph the function $$r(x)=\frac{1}{(x-2)^{2}}$$ by transforming the graph of \(y=1 / x^{2}\) (b) Use long division and factoring to show that the function $$s(x)=\frac{2 x^{2}+4 x+5}{x^{2}+2 x+1}$$ can be written as $$s(x)=2+\frac{3}{(x+1)^{2}}$$ Then graph \(s\) by transforming the graph of \(y=1 / x^{2}\) . (c) One of the following functions can be graphed by transforming the graph of \(y=1 / x^{2} ;\) the other cannot. Use transformations to graph the one that can be, and explain why this method doesn't work for the other one. $$p(x)=\frac{2-3 x^{2}}{x^{2}-4 x+4} \quad q(x)=\frac{12 x-3 x^{2}}{x^{2}-4 x+4}$$

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example 3(a). $$ P(x)=3 x^{3}-5 x^{2}-8 x-2 $$

How Many Real Zeros Can a Polynomial Have? Give examples of polynomials that have the following properties, or explain why it is impossible to find such a polynomial. (a) A polynomial of degree 3 that has no real zeros (b) A polynomial of degree 4 that has no real zeros (c) A polynomial of degree 3 that has three real zeros, only one of which is rational (d) A polynomial of degree 4 that has four real zeros, none of which is rational What must be true about the degree of a polynomial with integer coefficients if it has no real zeros?

\(41-58=\) Find all zeros of the polynomial. $$ P(x)=x^{3}-2 x^{2}+2 x-1 $$
See all solutions

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