Problem 63 - Algebra 2

Question

Simplify each expression. $$ \frac{3 y+1}{2 y-10}+\frac{1}{y^{2}-2 y-15} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{3y^2 + 10y + 5}{2(y-5)(y+3)}\).

Step 1: Factor the Denominators

Start by factoring the denominators of the fractions. For the first fraction, the denominator is \(2y-10\) which factors to \(2(y-5)\). For the second fraction, the denominator is \(y^2 - 2y - 15\), which factors to \((y-5)(y+3)\).

Step 2: Find a Common Denominator

Next, find a common denominator for the two fractions. The denominators are \(2(y-5)\) and \((y-5)(y+3)\). The least common denominator (LCD) is \(2(y-5)(y+3)\).

Step 3: Adjust Each Fraction to the LCD

Adjust the first fraction \(\frac{3y+1}{2(y-5)}\) to have the common denominator \(2(y-5)(y+3)\). Multiply the numerator and the denominator by \((y+3)\), resulting in \(\frac{(3y+1)(y+3)}{2(y-5)(y+3)}\). For the second fraction \(\frac{1}{(y-5)(y+3)}\), multiply the numerator and denominator by 2 to get \(\frac{2}{2(y-5)(y+3)}\).

Step 4: Combine the Fractions

Combine the fractions since they now have a common denominator. Add the numerators: \((3y+1)(y+3) + 2\), resulting in \[\frac{(3y+1)(y+3) + 2}{2(y-5)(y+3)}\].

Step 5: Simplify the Numerator

Expand and simplify the numerator. \((3y+1)(y+3)\) expands to \(3y^2 + 9y + y + 3\), which simplifies to \(3y^2 + 10y + 3\). Add 2 to get \(3y^2 + 10y + 5\).

Step 6: Final Simplified Expression

The final simplified expression is \[\frac{3y^2 + 10y + 5}{2(y-5)(y+3)}\].

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial step when simplifying algebraic expressions, particularly rational expressions. It involves expressing a polynomial as a product of its factors to simplify complex expressions.
For example, consider the polynomial in the second fraction’s denominator:

• Original polynomial: \(y^2 - 2y - 15\)
• This can be factored into: \((y-5)(y+3)\)

Just like this example, identifying factors helps break down complicated expressions into simpler building blocks. This process makes it easier to find common denominators and perform operations like addition and subtraction. Remember, recognizing patterns such as the difference of squares or trinomials can greatly help in quickly factoring polynomials.

Rational Expressions

Rational expressions are fractions where the numerator and/or the denominator are polynomials. To work with them effectively, we must understand how to manipulate these expressions.

• Look for opportunities to factor both the numerator and the denominator.
• Simplify them much like you would numerical fractions, by reducing common factors.

In the provided exercise: - The rational expression \( \frac{3y+1}{2(y-5)}\) can be simplified by factoring the denominator.Simplifying rational expressions involves factoring wherever possible and finding a suitable least common denominator to perform operations such as addition or subtraction. Always remember to factor completely to see the full extent of possible simplifications. It’s also crucial to ensure the expressions do not have undefined parts, like denominators equaling zero.

Least Common Denominator

When adding or subtracting rational expressions, finding the least common denominator (LCD) is essential. This allows you to combine the fractions efficiently.
In our example, rational expressions have the following denominators:- \(2(y-5)\) and- \((y-5)(y+3)\)The LCD is determined by taking each distinct factor with the highest power found in any denominator:

• Combine unique factors: \(2\), \(y-5\), and \(y+3\).
• Resulting LCD: \(2(y-5)(y+3)\).

Using the LCD allows the fractions to share a common denominator, simplifying the addition or subtraction process. Adjust each fraction accordingly by multiplying both the numerator and denominator by the necessary factors to achieve this common denominator.

Combine Fractions

Once we’ve found the least common denominator, combining fractions becomes straightforward. All we have to do is adjust each fraction so it has this common denominator.In the given problem, after we've modified:- First fraction: \( \frac{(3y+1)(y+3)}{2(y-5)(y+3)}\)- Second fraction: \( \frac{2}{2(y-5)(y+3)}\)The fractions now share the same denominator \(2(y-5)(y+3)\). We can add their numerators directly:

• Add numerators: \((3y+1)(y+3) + 2\)
• Keep the common denominator: \(2(y-5)(y+3)\)

The result will be a single rational expression. Always simplify the result if possible. In summary, aligning denominators is crucial in combining fractions, ensuring the operation is seamless and results in a single, simplified expression.