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Chapter 4: Problem 91

Find a polynomial equation with the given solutions. 0, \(-4\)

Short Answer

Expert verified
The polynomial is \(x^2 + 4x = 0\).

Step by step solution

01

Understand the given solutions

The given solutions are the roots of the polynomial equation, which means the polynomial will be equal to zero when evaluated at these values. In this exercise, the roots given are 0 and \(-4\).
02

Write down the factors corresponding to the roots

For each solution or root \(r\), the corresponding factor of the polynomial is \((x - r)\). If 0 is a root, the factor is \(x\). If \(-4\) is a root, the factor is \(x + 4\). This gives us our factorized polynomial as \((x)(x + 4)\).
03

Multiply the factors to find the polynomial

To find the polynomial equation, multiply the factors derived from the roots. Here, we multiply \(x\) and \(x + 4\): \[ x(x + 4) = x^2 + 4x \]. This is the polynomial in its standard form.

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Key Concepts

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

Roots of a Polynomial
The roots of a polynomial are the values of the variable where the polynomial equals zero. These are essentially the solutions to the polynomial equation. For example, if you have a polynomial equation, say \(x^2 + 4x = 0\), finding its roots means determining the values of \(x\) that satisfy this equation.

When you're given the roots, these values can be directly used to construct the polynomial.
  • If 0 is one of the roots, it means that when \(x = 0\), the polynomial equals zero.
  • Similarly, if \(-4\) is a root, then the polynomial is zero when \(x = -4\).
Understanding roots is crucial because it allows us to use them in reverse to build or verify polynomial equations. Given roots can be translated into factors, which then lead to the complete polynomial.
Factoring Polynomials
Factoring polynomials involves expressing the polynomial as a product of its factors. Each root of a polynomial is associated with a factor. For a given root \(r\), the corresponding factor is \((x - r)\).

For instance:
  • Given a root of 0, the factor is \(x\).
  • Given a root of \(-4\), the factor becomes \(x + 4\).
Factorization is an essential step in solving polynomials as it breaks down complex expressions into simpler multiplicative components.

After identifying the factors, the next logical step is multiplying them to form the polynomial expression. For instance, multiplying the factors \((x)(x + 4)\) results in the polynomial \(x^2 + 4x\).

This process allows us to move from a set of solutions (roots) to a polynomial equation that possesses those solutions.
Standard Form of a Polynomial
The standard form of a polynomial is its expanded version where all the terms are expressed as powers of \(x\) ordered from the highest power down to the constant term. For a polynomial to be in standard form, it must be organized in descending order of exponents.

In our example:
  • The factors \(x\) and \(x + 4\) were multiplied to get \(x^2 + 4x\).
  • This expanded version is \(x^2 + 4x\), which is now in standard form. Here, the highest degree term is \(x^2\), followed by \(x\).
Being comfortable with converting factored polynomials into their standard form is necessary because this form provides a clear view of the polynomial鈥檚 degree and facilitates further operations or analysis like addition, subtraction, or calculus-based computations with polynomials.

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