Problem 20 - Elementary Algebra

Question

Simplify \(2 a-5-\frac{a^{2}+2 a-1}{a+3}\)

Step-by-Step Solution

Answer

Question: Simplify the given expression: \(2a - 5 - \frac{a^2+2a-1}{a+3}\) Answer: \(\frac{a^2-a-14}{a+3}\)

Step 1: Identify the common denominator

In this expression, the fraction has a denominator of \(a+3\). We need to find a common denominator for all terms to combine the expressions. Since there are no other fractional terms in this expression, the common denominator is simply \(a+3\).

Step 2: Rewrite the terms with the common denominator

Next, we rewrite the terms so that each has a denominator of \(a+3\). For the non-fractional terms, multiply them by \(\frac{a+3}{a+3}\). \((2a) \cdot \frac{a+3}{a+3} - (5) \cdot \frac{a+3}{a+3} - \frac{a^{2}+2a-1}{a+3}\)

Step 3: Expand and simplify the numerator of each term

Now, we expand the numerator of each term and simplify it: \(\frac{2a(a+3)}{a+3} - \frac{5(a+3)}{a+3} - \frac{a^2+2a-1}{a+3}\) \(\frac{2a^2+6a}{a+3} - \frac{5a+15}{a+3} - \frac{a^2+2a-1}{a+3}\)

Step 4: Combine the terms with the common denominator

Since all terms share the same denominator of \(a+3\), we can now combine the numerators: \(\frac{2a^2+6a-5a-15-a^2-2a+1}{a+3}\)

Step 5: Simplify the numerator of the fraction

Finally, we simplify the combined numerator by adding the coefficients of like terms: \(\frac{2a^2-a^2+6a-5a-2a-15+1}{a+3}\) \(\frac{a^2-a-14}{a+3}\) The simplified expression is \(\frac{a^2-a-14}{a+3}\).

Key Concepts

Common Denominator

When we work with expressions involving fractions, the concept of a common denominator is essential. It allows us to manipulate the expression by ensuring all fractions involved share the same base. This shared denominator makes it possible to conveniently combine or compare terms.

In our exercise, the expression consists of a polynomial fraction with a denominator of \(a + 3\). Although other terms in the expression are not naturally fractions, we treat them as such by rewriting them with a common denominator. Thus, every term in the expression, fractional or not, is rewritten to have \(a + 3\) as its denominator. This approach facilitates the combination of terms and is crucial for simplifying expressions with mixed operations.

Polynomial Fractions

Polynomial fractions are fractions where the numerator, the denominator, or both, contain polynomials. These types of expressions require careful treatment due to the complex nature of polynomials.

In the given problem, the expression \( \frac{a^2 + 2a - 1}{a + 3} \) is a polynomial fraction. The numerator is a quadratic polynomial, whereas the denominator is linear. Working with polynomial fractions involves:

• Identifying polynomials in both numerators and denominators.
• Finding common denominators if needed for combining operations.
• Simplifying complex numerators for easier manipulation.

The ultimate goal is to simplify the expression, making it easily manageable and understandable.

Numerator Simplification

Numerator simplification is the process of combining like terms and reducing parts of the numerator to a simpler form. This process often involves expanding multiplied expressions and combining terms with similar variables or constants.

In this exercise, after adjusting all terms to have the common denominator \(a + 3\), we need to simplify the numerators. The expression transforms to:

• \(2a^2 + 6a\)
• \(-5a - 15\)
• \(-a^2 - 2a + 1\)

These parts are all combined over the common denominator. We group like terms and operate on them, like treating \(2a^2 - a^2\) to make it \(a^2\). We also add and subtract linear terms accurately, and handle constants effectively. This process gives us a significantly simplified numerator, resulting in the final simplified expression of \( \frac{a^2 - a - 14}{a + 3} \). Simplifying numerators not only clears up complexity but also makes further algebraic manipulation much more intuitive.