Problem 37 - Introductory Algebra for College Students

Question

add or subtract as indicated. Simplify the result, if possible. $$\frac{6 y^{2}+y}{2 y^{2}-9 y+9}-\frac{2 y+9}{2 y^{2}-9 y+9}-\frac{4 y-3}{2 y^{2}-9 y+9}$$

Step-by-Step Solution

Answer

Thus, the simplified result of the given expression is \( \frac{6 y^{2} - 5 y - 6}{2 y^{2}-9 y+9} \).

Step 1: Combine Numerators

Since all of the fractions have the same denominator, you can simply add and subtract the numerators. Do this by subtracting the second and third numerators from the first numerator, carefully distributing the negative sign: \( (6 y^{2}+y) - (2 y+9) - (4 y-3) \)

Step 2: Simplify Numerator

Next, combine like terms: \( 6 y^{2} + y - 2 y - 9 - 4 y + 3 \). This simplifies to \( 6 y^{2} - 5 y - 6 \)

Step 3: Rewrite as Fraction

Now rewrite the simplified numerator over the common denominator to give the result as a single fraction: \( \frac{6 y^{2} - 5 y - 6}{2 y^{2}-9 y+9} \).

Key Concepts

Rational Expressions

Rational expressions are a type of algebraic expression that feature a ratio of two polynomials. In simpler terms, they are fractions where both the numerator (top part) and the denominator (bottom part) are polynomials. Just like fractions, rational expressions can be manipulated by addition, subtraction, multiplication, and division.

Understanding these expressions is crucial because they frequently appear in algebra and calculus problems. Simplifying rational expressions can sometimes be a complex task, especially when the polynomials have higher degrees or share multiple common factors. To master this concept:

• Always check if there is scope to factorize the numerator or the denominator.
• Look for common factors in the polynomials that can be canceled out.
• Be cautious with every operation, ensuring each step follows algebraic rules.

Adding and Subtracting Fractions

Adding and subtracting fractions is a fundamental skill in algebra. The main rule is that both fractions must have a common denominator before they can be added or subtracted. When working with rational expressions, this principle translates similarly:

• Ensure all expressions have the same denominator before proceeding.
• Carefully combine the numerators, distributing any negative signs appropriately.
• After adding or subtracting, always look for opportunities to simplify the resulting expression.

In the given exercise, you'll notice all fractions shared the same denominator, making the combination of numerators straightforward. Be mindful of negative signs during subtraction, as they affect each term in the numerator.

Polynomial Simplification

Polynomial simplification is the process of reducing a polynomial to its simplest form. This involves combining like terms and factoring where possible. Like terms in a polynomial are terms that have the same variable raised to the same power. Simplification is critical when working with algebraic expressions, as it makes other operations more manageable.

To effectively simplify a polynomial, use these tips:

• Identify and group similar terms, such as all terms with the same degree (e.g., all terms with a 'y' or 'y²').
• Carefully perform arithmetic operations with each group.
• Check if further factoring can simplify your expression further or if any terms can be canceled through division.

In the exercise provided, you first combine like terms in the numerator to reduce the algebraic fraction effectively. This leads to a more concise expression, which is often easier to interpret and work with.