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Factoring polynomials involves breaking down a complex expression into simpler factors. These factors multiply together to give the original expression. It's like breaking a number into its prime factors, but for algebraic expressions. When you have quadratic polynomials like the one in our example, a variety of patterns can help. One commonly used pattern is the difference of squares.

Why factor polynomials? It simplifies expressions and makes them easier to work with, especially in algebra division. If you're given the polynomial like our expression:

• First, identify any recognizable patterns (e.g., sums or differences of squares).
• Apply known formulas to factorize.
• Recheck your factored terms by expanding to ensure accuracy.

In this exercise, the original expression was factored into \((5x + 4)(5x - 4)\), simplifying further operations.

The difference of squares is a special type of polynomial that takes the form of \(a^2 - b^2\). It can be rewritten as \((a + b)(a - b)\). This concept is particularly handy because expressing complex polynomial differences in simpler products can be done swiftly using this formula.

For example, in our exercise, \(25x^2 - 16\) is expressed as:

• The square of \(5x\), which is \((5x)^2\), giving \(25x^2\).
• The square of \(4\), giving \(16\) or \(4^2\).

Hence, matching the difference of squares pattern, it simplifies to \((5x + 4)(5x - 4)\). This simplification paves the way to easily reduce and solve polynomials when dividing by another expression, as \((a - b)\) or \((a + b)\) often cancels out a corresponding term in the denominator.

Simplifying rational expressions is crucial for understanding and solving algebra problems efficiently. It involves reducing complex fractions into simpler forms while ensuring that the basic value remains unchanged. This is made feasible by eliminating common factors from both the numerator and the denominator.

To simplify a rational expression:

• Identify common factors in the numerator and denominator.
• Factor these expressions completely.
• Cancel out common factors, if present, ensuring you're not dividing by zero.

In our exercise, the factored form of the numerator \((5x + 4)(5x - 4)\) was divided by \(5x - 4\) in the denominator. Canceling the \(5x - 4\) factor simplifies the division, leaving a neat expression of \(5x + 4\). This process reveals the final simplified form, often a much less cumbersome expression to work with. Always ensure to include conditions like \(x eq 0.8\) to avoid zero division in rational expressions.