Problem 45 - Algebra A Combined Function

Question

Simplify each expression. $$ \frac{x^{2}+7 x+10}{x^{2}-3 x-10} $$

Step-by-Step Solution

Answer

\(\frac{x+5}{x-5}\)

Step 1: Identify Common Factors

Examine both the numerator and the denominator to see if they can be factored. The numerator is \(x^2 + 7x + 10\) and the denominator is \(x^2 - 3x - 10\).

Step 2: Factor the Numerator

Factor the quadratic \(x^2 + 7x + 10\). Find two numbers that multiply to 10 and add to 7, which are 5 and 2. Thus, \(x^2 + 7x + 10 = (x + 5)(x + 2)\).

Step 3: Factor the Denominator

Factor the quadratic \(x^2 - 3x - 10\). Find two numbers that multiply to -10 and add to -3, which are -5 and 2. Thus, \(x^2 - 3x - 10 = (x - 5)(x + 2)\).

Step 4: Simplify the Fraction

The expression is now \(\frac{(x+5)(x+2)}{(x-5)(x+2)}\). Notice that \((x + 2)\) is a common factor in both the numerator and the denominator, and can thus be canceled out. This yields \(\frac{x+5}{x-5}\).

Step 5: Verify Simplification

Double-check that no additional factors can be canceled and note that \(x+5\) and \(x-5\) have no common factors. The simplification process is complete with the final form being the correct result.

Key Concepts

Factoring Quadratics

Factoring quadratics is a crucial step in simplifying algebraic expressions involving quadratic expressions. Quadratics are polynomials of degree two, and factoring them involves breaking them down into simpler, linear factors. The general form of a quadratic expression is \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are coefficients.

When factoring, we look for two numbers that multiply to \( ac \) and add to \( b \). In our exercise, the numerator \( x^2 + 7x + 10 \) factors easily by finding two numbers that multiply to 10 and add up to 7, which are 5 and 2. Therefore, we can write it as \( (x+5)(x+2) \).

For the denominator, \( x^2 - 3x - 10 \), the challenge is to find two numbers that multiply to -10 (the product of \( a \) and \( c \) in this case) and sum to -3. The numbers are -5 and 2, leading us to factor the expression as \( (x-5)(x+2) \). With practice, factoring becomes more intuitive, allowing students to swiftly simplify expressions.

Polynomial Division

Polynomial division is a method used to simplify expressions by dividing one polynomial by another. In our exercise, rather than performing long division, which can be quite complex and time-consuming, we simplify by factoring both the numerator and the denominator.

The divided expression forms a fraction where the numerator and denominator have been factored into \( (x+5)(x+2) \) and \( (x-5)(x+2) \), respectively. This method, known as simplifying by factoring, takes advantage of common factors to simplify the polynomial fraction to a simpler form \( \frac{x+5}{x-5} \).

This step is crucial in polynomial division as it allows us to understand the behavior of the polynomial when at certain values, specifically where the denominator equals zero, which might lead to points of discontinuity in functions.

Fraction Simplification

Fraction simplification involves reducing a fraction to its simplest form by canceling out common factors from the numerator and the denominator. This process makes working with the fraction easier and helps reveal the core expression without redundant components.

In our exercise, after factoring the quadratic expressions, the fraction becomes \( \frac{(x+5)(x+2)}{(x-5)(x+2)} \). The factor \( (x+2) \) is found in both the numerator and the denominator, so it can be cancelled out, leaving us with \( \frac{x+5}{x-5} \).

It's vital to emphasize that fraction simplification does not change the value of the expression, but rather presents it in more manageable terms. Always remember to check for any remaining common factors to ensure you've simplified thoroughly.