91Ó°ÊÓ

Determining roots of a polynomial equation means finding the values of the variable that make the equation true. In the example given, we have two polynomial equations:

• Equation 1: \(x^3 + x^2 - x - 1 = 0\)
• Equation 2: \(x^3 + 2x^2 - 4x - 8 = 0\)

By 'root', we mean the value or values of x for which these equations equal zero. To determine if these roots are positive or negative, you can test simple values like 1, -1, 2, and -2. In the exercise solution, substituting x = 1 into Equation 1 confirms that \(1^3 + 1^2 - 1 - 1 = 0\)br>Similarly, substituting x = 2 into Equation 2 confirms that \(2^3 + 2*2^2 - 4*2 - 8 = 0\). By solving these polynomials, we know these are roots of the respective equations. Understanding this allows us to conclude the possible values that make the polynomial equation true.

Polynomial equations involve terms with variables raised to powers, all set equal to zero. A polynomial is just an algebraic expression with one or more terms. For example, \(x^3 + x^2 - x - 1\) is a polynomial with four terms: \(x^3, x^2, -x, \text{and} -1\). When you solve a polynomial equation, you're trying to find the roots, in other words, the values of x that make the equation zero. For the given equations, \(x^3+x^2-x-1=0\)and \(x^3+2x^2-4x-8=0\), the roots were found by testing simple integers. The importance of understanding polynomial equations lies in breaking down complex expressions into simpler components to find solutions.

Determining whether a root is positive or negative is essential for understanding the nature of the solutions. In the example exercise, testing for positive and negative roots helps us determine whether x is positive or negative. Testing \(x = 1\) in Equation 1 shows \(1^3 + 1^2 - 1 - 1 = 0\), confirming x can be positive. Similarly, for Equation 2, testing \(x = 2\) shows \(2^3 + 2*2^2 - 4*2 - 8 = 0\), confirming another positive root. So, by testing values, we not only find the roots but also verify their nature, whether positive or negative. This process helps in visualizing where the roots lie on the number line relative to zero, enhancing our understanding of polynomial behavior.