Problem 31 - Algebra 1: Concepts and Skills

Question

FACTORING AFTER ADDING OR SUBTRACTING. Simplify the expression. $$ \frac{y^{2}}{y^{2}-3 y-28}-\frac{12-y}{y^{2}-3 y-28} $$

Step-by-Step Solution

Answer

\[\frac{y-3}{y-7}\]

Step 1: Combining Numerators

Given two fractional expressions having the same denominator \( y^{2}-3 y-28 \), concatenate their numerators with the operation in between. This results in \[\frac{y^{2}}{y^{2}-3 y-28}-\frac{12-y}{y^{2}-3 y-28} = \frac{y^{2}-(12-y)}{y^{2}-3 y-28}.\]

Step 2: Simplify Numerator

Next, simplify the numerator by applying the distributive property to the expression \(y^{2}-(12-y)\) to get \[\frac{y^{2}-(12-y)}{y^{2}-3 y-28} = \frac{y^{2}-12+y}{y^{2}-3 y-28}.\]

Step 3: Combine Like Terms

After simplification, combine like terms \(y^{2}\) and \(y\) in the numerator to give \[\frac{y^{2}-12+y}{y^{2}-3 y-28} = \frac{y^{2}+y-12}{y^{2}-3 y-28}.\]

Step 4: Factor

Since the numerator and the denominator are both quadratic expressions, look for factors in order to simplify the expression further. The numerator factors as \((y+4)(y-3)\) and the denominator as \((y+4)(y-7)\). So the expression simplifies to \[\frac{y^{2}+y-12}{y^{2}-3 y-28} = \frac{(y+4)(y-3)}{(y+4)(y-7)}.\]

Step 5: Simplify

Since \((y+4)\) is a common factor to both the numerator and the denominator it can be simplified to give \[\frac{(y+4)(y-3)}{(y+4)(y-7)} = \frac{y-3}{y-7}.\] This is the simplified version of the original expression.

Key Concepts

Quadratic Expressions

Quadratic expressions are polynomials of degree two. They take the general form: \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( x \) is the variable. These expressions frequently appear in algebra.
Quadratic expressions can be factored into two binomials. For example, the expression \( y^2 + y - 12 \) can be broken down to \( (y+4)(y-3) \).
To factor, look for two numbers that multiply to give the constant term \( c \) and add to give the middle coefficient \( b \). This process allows us to break down complex quadratics and find their roots or simplify fraction operations more easily.
Understanding factors is crucial because it helps not only in solving fractions but also in graphing parabolas and solving quadratic equations. Recognizing patterns in quadratic expressions can become intuitive with practice.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form, meaning you cancel out any common factors in the numerator and the denominator. This process can make calculations and expressions much easier to handle.
In this exercise, we combined fractions that had the same denominator. This made the simplification possible.
To simplify:

• Ensure that the fractions have a common denominator.
• Combine the numerators appropriately, respecting the operation between the fractions.
• Factor both the numerator and the denominator if they're quadratic or polynomials.
• Cancel out any common factors from the numerator and the denominator.

This helps in achieving a simpler form that's more convenient to work with, especially in subsequent mathematical operations.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying algebraic expressions to solve problems more effectively. It's like playing with blocks, where the rules of operations guide each move.
In this problem:

• We first handled the subtraction between the fractions by merging the numerators. This is possible since they had the same denominator.
• Next, we manipulated terms in the numerators. Combining like terms helps in reducing complexity and clarifying the expression.
• Then came factoring, where breaking down the quadratic expressions into products of binomials facilitated further simplification.

These steps showcase vital algebraic skills, such as the distributive property \( a(b+c) = ab + ac \), and combining like terms, which are foundational for tackling more complex algebraic expressions.
Knowing how to manipulate algebraic expressions is essential in problem-solving across various math topics, enabling clearer thinking and more efficient solutions.