Question

Write the rational expression in simplest form.\(\frac{y^{2}-7 y+12}{y^{2}+3 y-18}\)

Short answer

The simplified form of the given rational expression is \((y - 4)/(y + 6)\).

Step-by-step solution

Factor the numerator

The numerator is of the form \(a^2 - b^2\) which is a quadratic trinomial. Factoring the numerator \(y^2 - 7y + 12\) we will get \((y - 3)(y - 4)\).

Factor the denominator

The denominator is also a quadratic trinomial. Factoring the denominator \(y^2 + 3y - 18\) we will get \((y - 3)(y + 6)\).

Simplify the expression

Now we simplify the expression by dividing the both the numerator and the denominator by the common term. So the rational expression \((y - 3)(y - 4) / (y - 3)(y + 6)\) simplifies to \((y - 4)/(y + 6)\).

Key concepts

Factoring Quadratic Trinomials

Understanding how to factor quadratic trinomials is a fundamental skill in algebra, as these are expressions that appear frequently. A quadratic trinomial is typically in the form of ax^2 + bx + c, where a, b, and c are constants, and the highest power of the variable x is 2. The goal of factoring is to break the quadratic trinomial into a product of two binomials.

To factor a quadratic trinomial, one must find two numbers that not only multiply to give the constant term c but also add up to the coefficient of the middle term b. This might sound daunting, but with practice, patterns begin to emerge. For the expression \(y^2 - 7y + 12\), we look for two numbers that multiply to 12 (the constant term) and add up to -7 (the coefficient of the middle term). These two numbers are -3 and -4, giving us the factorized form \(y - 3)(y - 4)\).

It's essential to check that the product of the binomials brings you back to the original trinomial to ensure the accuracy of the factoring process.

Common Factors in Rational Expressions

When working with rational expressions, identifying common factors in the numerator and denominator can greatly simplify the expression. A common factor is a term that is present in both the numerator and the denominator of a fraction, which can be canceled out. This process is like reducing a numerical fraction to its lowest terms by dividing both the numerator and denominator by their greatest common divisor.

For example, in the rational expression \( \frac{y^2-7y+12}{y^2+3y-18} \), after factoring, we obtain \( \frac{(y - 3)(y - 4)}{(y - 3)(y + 6)} \). Here, \(y - 3\) is a common factor. We can eliminate this term from both the numerator and the denominator, as it is essentially dividing by itself, which simplifies to 1. Remember to be cautious that you’re not canceling out terms incorrectly; they must be factors, not terms that are just part of a sum or difference.

Algebraic Simplification

Algebraic simplification involves reducing expressions to their simplest form without changing their value. This process often includes several steps such as distributing, combining like terms, and canceling common factors. When simplifying rational expressions, it's crucial to recognize opportunities to factor and reduce.

After identifying common factors in the previous step, we proceed to the simplification. Continuing with our example, once the common factor \(y - 3\) is eliminated, the original complex fraction becomes \( \frac{y - 4}{y + 6} \). This is now the simplified form of the original rational expression, as it can't be reduced any further. Simplification is not only about making an expression shorter; it is also about making it more comprehensible, which aids in solving equations or evaluating expressions. As such, being thorough in your simplification process ensures clarity and correctness in your algebraic endeavors.