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Chapter 4: Problem 94

Find a polynomial equation with the given solutions. \(\pm 2\)

Short Answer

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

Step by step solution

01

Identify Given Solutions

The problem provides the solutions as \( \pm 2 \). This means the roots of the equation are \( +2 \) and \( -2 \).
02

Translate Solutions into Factors

For each solution \( x = 2 \) and \( x = -2 \), we can write the corresponding factors of the polynomial equation as \( (x - 2) \) and \( (x + 2) \).
03

Write the Polynomial Equation

The polynomial equation is formed by multiplying the factors found in the previous step. We have the equation: \( (x - 2)(x + 2) \).
04

Expand the Factors

Now, expand the expression \( (x - 2)(x + 2) \) using the difference of squares formula: \( a^2 - b^2 = (a - b)(a + b) \). Here, set \( a = x \) and \( b = 2 \), so the product becomes \( x^2 - 4 \).
05

State the Polynomial Equation

The polynomial equation with solutions \( \pm 2 \) is \( x^2 - 4 = 0 \).

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Key Concepts

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

Roots of a Polynomial
When solving a polynomial equation, a crucial step is finding the "roots" or "solutions". These are values of the variable (commonly denoted as \( x \)) that make the entire equation equal to zero. A solution to the polynomial equation \( f(x) = 0 \) is a root if substituting it into the function \( f(x) \) results in zero. For example, if the equation is \( x^2 - 4 = 0 \), substituting \( x = 2 \) or \( x = -2 \) into the equation will yield zero. Thus, \( 2 \) and \( -2 \) are the roots.
  • Understanding roots helps determine solutions that satisfy the polynomial equations.
  • They play a significant role in factoring, as each root corresponds to a factor of the polynomial.
Understanding the roots is essential when constructing a polynomial equation from given solutions as it guides the formulation of its factors.
Difference of Squares
The difference of squares is a particular algebraic structure expressing the form \( a^2 - b^2 \). This can be factored into the product of two binomials: \( (a - b)(a + b) \). This formula is particularly useful because it simplifies the process of multiplying polynomial expressions or factoring them back to their roots.This concept appears when working with a polynomial like \( x^2 - 4 \). Here, \( 4 \) can be seen as \( 2^2 \), so it fits the difference of squares pattern \( x^2 - 2^2 \). Breaking it down:
  • Let \( a = x \) and \( b = 2 \).
  • Applying the difference of squares gives \( (x - 2)(x + 2) \).
Remember, recognizing difference of squares can streamline both factoring and expanding polynomial expressions, a pivotal technique in algebra.
Factoring Polynomials
Factoring is a mathematical process of breaking down a complex expression into simpler factors, which when multiplied together give back the original expression. With polynomials, factoring involves rewriting them as a product of their roots or simpler polynomials.Consider the polynomial \( x^2 - 4 \). This polynomial can be factored using its roots \( 2 \) and \( -2 \). It involves expressing these roots as factors:
  • Identify the roots: \( x = 2 \) and \( x = -2 \).
  • Translate these roots into factors: \( (x - 2) \) and \( (x + 2) \).
  • The factorization of \( x^2 - 4 \) becomes \( (x - 2)(x + 2) \).
By understanding factoring, one can effectively simplify polynomials, solve them, and explore their properties. Being proficient in factoring provides a foundation for mastering more advanced algebraic concepts.

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Most popular questions from this chapter

A manufacturer has determined that the cost in dollars of producing bicycles is given by \(C(x)=0.5 x^{2}-x+6200\), where \(x\) represents the number of bicycles produced weekly. Determine the average cost of producing 50, 100, and 150 bicycles per week.

A boat can average 10 miles per hour in still water. On a trip downriver, the boat was able to travel \(7.5\) miles with the current. On the return trip, the boat was only able to travel \(4.5\) miles in the same amount of time against the current. What was the speed of the current?

Find a function with the given roots. \(-1,0,3\)

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