Problem 100 - Elementary Algebra

Question

Reduce \(\frac{y^{2}-y-6}{y-3}\)

Step-by-Step Solution

Answer

Answer: The reduced expression is \(y+2\).

Step 1: Factor the numerator

First, we need to factor the numerator. The expression \(y^{2}-y-6\) can be factored into two binomials. To do this, find two numbers that multiply to -6 and add up to -1. The numbers are -3 and 2. So the factored form of the numerator is \((y-3)(y+2)\).

Step 2: Simplify the expression

Now that we have factored the numerator, we can simplify the expression by canceling out the common factors in the numerator and denominator. As both the numerator and denominator contain the factor \((y-3)\), we will cancel it out, which gives us the reduced expression: $$\frac{(y-3)(y+2)}{y-3} \Rightarrow \frac{\cancel{(y-3)}(y+2)}{\cancel{y-3}} = y+2$$ So, the reduced expression is \(y+2\).

Key Concepts

Factoring Polynomials

Factoring polynomials is a key skill in algebra that involves breaking down a polynomial into simpler components, called factors. This makes polynomials easier to work with, particularly when simplifying expressions or solving equations. To factor a quadratic polynomial like \( y^2 - y - 6 \), start by identifying two numbers that multiply to the constant term, \(-6\), and add up to the coefficient of the linear term, \(-1\). This method, often referred to as "factoring by grouping," is very common for quadratics. For \( y^2 - y - 6 \), the numbers \(-3\) and \(2\) satisfy these conditions. Therefore, the expression can be rewritten as \((y - 3)(y + 2)\).

Factoring polynomials requires practice, and it can become intuitive over time. Always check your work by expanding the factors to ensure they give the original polynomial.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. They form a crucial part of algebra because they allow the manipulation and simplification of expressions that may initially seem complex. Consider the given expression \(\frac{y^2-y-6}{y-3}\), where the numerator is a polynomial \( y^2 - y - 6 \), and \( y - 3 \) is the denominator. These types of expressions become much more manageable once you learn to factor and cancel common factors.

The goal is usually to reduce these expressions to the simplest form, simplifying calculations and further analysis. It's important to remember that any solution must consider restrictions on the variable, such as values that make the denominator zero. For example, in our original expression, \( y \) cannot be \(3\), because this would make the denominator zero, leading to an undefined expression.

Simplifying Expressions

Simplifying expressions is about making them as readable and easy to manage as possible. It involves reducing expressions to their simplest forms, ensuring that no further cancellation is possible. In the example \(\frac{y^2-y-6}{y-3}\), once we factor the numerator as \((y-3)(y+2)\), we can simplify by canceling the common factor \( y - 3 \) from both the numerator and the denominator.

The simplified expression \( y + 2 \) is much easier to interpret and use in further mathematical manipulations.

• First, completely factor both the numerator and the denominator.
• Next, identify and cancel any common factors.
• Always verify that what remains is in its simplest form.

Simplifying can also involve other operations, such as combining like terms or using identities to re-arrange terms. Clear and simple expressions are often the key to solving more complex algebraic problems efficiently.