91Ó°ÊÓ

Factoring is a crucial mathematical process that involves breaking down expressions into simpler terms, or "factors," which, when multiplied together, produce the original expression. This technique is especially useful for solving polynomial equations. In the given exercise, both terms contain common factors

• \((x+1)^{2}\) and
• \((2x-1)^{3}\).

The process begins by identifying these common factors. This allows us to simplify the equation greatly, presenting it in a more workable form.
The expression after factoring out the common terms becomes:\[(x+1)^{2}(2x-1)^{3} \left[ 3(2x-1) + 8(x+1) \right] = 0\]Factoring is particularly important because it reduces complex expressions and makes them easier to analyze and solve. Once the expression is factored, each factor can be handled separately, as individual equations or inequalities.
In summary, factoring serves as a powerful stepping stone for finding solutions in both polynomial equations and inequalities.

Inequalities involve expressions that use signs like \(<\), \(>\), \(\leq\), or \(\geq\), depicting that one side is not equal but either smaller or larger than the other. In the problem presented, we deal with a polynomial inequality:

• \[(x+1)^{2}(2x-1)^{3}(14x+5) \geq 0\].

Solving this inequality means determining the sets of values for \(x\) that make this statement true.
A strategic method is performing "sign analysis." This involves testing the sign (positive or negative) of the entire product over intervals defined by critical points (roots from the equation part). We then identify where the entire expression remains non-negative.
For the intervals:

• \((x < -1)\)
• \((-1 < x < -\frac{5}{14})\)
• \((-\frac{5}{14} < x < \frac{1}{2})\)
• \((x > \frac{1}{2})\)

analysis of each segment reveals where the inequality holds.
As a result, the solution set is \[(-\infty, -\frac{5}{14}] \cup [\frac{1}{2}, \infty)\],indicating all the intervals \(x\) can take for the compound product to be zero or positive.

Finding the roots of an equation is an essential task in solving polynomial equations. Roots are specific values of \(x\) that satisfy the equation, making it equal to zero. For the equation \[(x+1)^{2}(2x-1)^{3}(14x+5) = 0\],we identify roots by setting each factor to zero one at a time.

• The factor \((x+1)^{2} = 0\) gives us the root \(x = -1\).
• The factor \((2x-1)^{3} = 0\) leads to \(x = \frac{1}{2}\).
• Similarly, the factor \(14x + 5 = 0\) results in \(x = -\frac{5}{14}\).

Each root tells us an essential detail about where the polynomial crosses the x-axis. These places are also crucial when analyzing inequalities, as transitions between positive and negative regions occur at these points.
Understanding where the equation is zero helps in comprehending the behavior of the polynomial overall. It is pivotal for analyzing the feasibility and range of solutions for related inequalities.