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Factoring polynomials is a method of expressing a polynomial as a product of its factors (reduced polynomials), which is useful in solving equations and simplifying expressions. When you factor a polynomial, you're essentially breaking it down into simpler polynomials that, when multiplied together, give the original polynomial.

When you encounter a polynomial, start by identifying its degree and checking for common factors among its terms. You can also employ specific strategies based on its form, such as the difference of squares or the difference of cubes.

• Finding the greatest common factor (GCF) can often simplify the process.
• Look for recognisable patterns that suggest using specific factoring techniques.

In our example, we factor both the numerator and the denominator of a rational expression to simplify it. The numerator is a straightforward case of the difference of squares.

The difference of squares is a specific polynomial form: the subtraction of one squared term from another. The formula for factoring a difference of squares is:
\[ a^2 - b^2 = (a + b)(a - b) \]
This formula shows that a difference of squares can always be factored into the product of two binomial expressions, one with a sum and the other with a difference.

• This technique is very useful because it helps break down more complex polynomial expressions.
• Recognizing that both terms are perfect squares is key to applying this technique.

Applying this to our example, the numerator \( y^2 - 25 \) can be rewritten as \( (y + 5)(y - 5) \) since both \( y^2 \) and 25 are perfect squares, and the expression fits the difference of squares form.

The difference of cubes is another special case that involves the subtracting of two cubed terms. The general formula for factoring a difference of cubes is:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
This formula indicates that the expression can be broken down into the product of a binomial and a trinomial.

• The binomial \((a - b)\) reflects the simple difference.
• The trinomial \((a^2 + ab + b^2)\) includes all cross terms, ensuring full factoring coverage.

In our example, the expression \( y^3 - 125 \) fits the difference of cubes form since both terms are cubes: \( y^3 \) and \( 125 = 5^3 \). Using the formula, it factors into \((y - 5)(y^2 + 5y + 25)\). By recognizing this pattern, the simplification of the expression becomes more manageable, especially after cancelling common factors in the rational expression.