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Factoring quadratics is a key skill in simplifying algebraic expressions. It involves breaking down a quadratic polynomial into multiplied binomial expressions. A standard quadratic equation is written as \( ax^2 + bx + c \). To factor, you need to find two numbers that multiply to \( ac \) and add to \( b \). In the example, we have \( 2x^2 + 11x - 6 \), where \( a = 2 \), \( b = 11 \), and \( c = -6 \).

First, look for pairs of factors of \( -12 \) (since \( 2 \times -6 = -12 \)) that sum up to \( 11 \). These numbers are \( 12 \) and \( -1 \). Then, rewrite the middle term using these as \( 2x^2 + 12x - x - 6 \).

Next, factor by grouping: \( 2x(x + 6) - 1(x + 6) \). Now, factor out the common binomial factor \((x + 6)\), leading to \((2x - 1)(x + 6)\). Thus, through factoring, we have transformed the quadratic into a product of two binomials. Factoring is crucial as it helps reveal common factors with the denominator for simplification.

Simplifying algebraic expressions involves reducing an expression to its simplest form without changing its value. The goal is to make it easier to work with or understand. By using techniques like combining like terms, factoring, and cancelling out common terms, you can simplify complex-looking expressions.

In our example, after factoring, the expression \( (2x - 1)(x + 6) \) is divided by \( (x + 6) \). Notice that \( x + 6 \) is a common factor in both the numerator and the denominator. Because they are identical, they can be canceled out.

This process simplifies the fraction to just \( 2x - 1 \). Simplifying thus helps in reducing expressions to forms that are easier for calculation purposes or further algebraic manipulation. Always remember: only terms that multiply both the top and bottom of the fraction can cancel each other out.

Rational expressions are fractions where the numerator and the denominator are polynomials. The rules of fractions apply, which means simplifying them involves factoring and canceling common factors.

For example, consider the rational expression \( \frac{2x^2 + 11x - 6}{x+6} \). Here, both the numerator and denominator are polynomial expressions. In order to simplify, one often first factors the polynomials, if possible.

Once factored, as shown with \( (2x - 1)(x + 6) \) in the numerator, you check for common terms with the denominator. This is essential as it ensures the expression is in its lowest terms, and does not change the expression's value. After simplifying, the fraction results in \( 2x - 1 \).

Working with rational expressions often involves checking the domain to ensure no division by zero occurs, highlighting the importance of understanding every part of the expression and its factors.