91Ó°ÊÓ

When solving quadratic equations, finding the polynomial roots is crucial as they reveal the values of the variable that make the entire expression equal to zero. In the provided exercise, the equation is presented in a factored form:

• Factors are
  • \(x + 3\)
  • \(x^2 - 4\)

Each factor is individually set to zero to find its roots. For the first factor, \(x + 3\), this gives \(x = -3\).
Setting the second factor, \(x^2 - 4\), equal to zero involves a further step. Solving quadratic equations typically means finding such roots, and for a polynomial in a factored form like this, you equate each factor to zero. These solutions are the roots, which are \(x = -3\), \(x = 2\), and \(x = -2\).
Using roots in this context is straightforward once factoring is complete, but understanding how to find and verify them improves comprehension.

Factoring techniques are essential tools in solving quadratic equations and simplifying polynomials. In the given exercise, the equation was already partially factored as \((x + 3)(x^2 - 4) = 0\). This approach makes it easier to find solutions as the equation is broken down.

• Factor by Grouping: If an expression isn't straightforward to factor, sometimes breaking it into smaller groups can lead to factorable sub-expressions.
• Common Factor: Look for terms that share common factors, which can be factored out first.

In this specific exercise, the second term \(x^2 - 4\) is identified as a difference of squares, which can be factored further. Mastery of these factoring techniques simplifies the process of solving for roots, as seen in the step-by-step solution provided.

The difference of squares is a special factoring technique useful for quickly solving expressions of the form \(a^2 - b^2\). Such expressions can be factored into \((a - b)(a + b)\). Recognizing expressions as a difference of squares allows for faster solutions.
In the exercise, \(x^2 - 4\) is a classic example, which can be rewritten using this technique:

• \(x^2 - 4 = (x - 2)(x + 2)\)

This shows how a seemingly complex quadratic term can break down into two linear factors. These are easier to solve for roots, leading to solutions \(x = 2\) and \(x = -2\).
The effectiveness of this approach in finding polynomial roots makes it a staple in algebraic problem-solving.