Problem 48 - Algebra 1: Concepts and Skills

Question

Simplify \(\frac{24 y^{2}+24}{8 y-3}-\frac{73 y}{8 y-3}\). A. \(\frac{(8 y+3)(3 y+8)}{8 y-3}\) B. \(\frac{3 y-8}{(8 y-3)^{2}}\) C. \(\frac{3 y-8}{8 y-3}\) D. \(3 y-8\)

Step-by-Step Solution

Answer

The simplified form of the given expression is \(3y-8\). So, the correct answer is D. \(3y-8\).

Step 1: Identify Denominator

Firstly, note that both fractions have the same denominator, \(8y-3\). Therefore, we can simplify the overall fraction by combining the two numerators: \(24y^{2} + 24 - 73y\)

Step 2: Simplify Numerator

Rearrange the terms in the numerator to make them easier to deal with: \(24y^{2} - 73y + 24\).

Step 3: Factor Numerator

Factorize the terms in the numerator. This gives us: \(8y-3)(3y-8)\)

Step 4: Simplify Fraction

Now the fraction can be written as: \(\frac{(8y-3)(3y-8)}{8y-3}\). We can see that \(8y-3\) is present in both numerator and denominator. We cancel out this term.

Step 5: Final Simplification

After cancelling out, we are left with \(3y-8\). So, the final simplified form is \(3y-8\)

Key Concepts

Polynomials

Polynomials are mathematical expressions consisting of variables and coefficients, involving the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. They are important in algebra because they can model a wide variety of problems and situations.

• Terms: Each combination of a variable raised to an exponent and multiplied by a coefficient is called a "term." For example, in the expression \(24y^2 - 73y + 24\), the terms are \(24y^2\), \(-73y\), and \(24\).
• Degree: The degree of a polynomial is the highest power of the variable in the polynomial. The degree tells you the behavior of the polynomial function as the variable values get very large.
• Factoring Polynomials: Factoring involves writing the polynomial as a product of its factors. In the solution, the polynomial \(24y^2 - 73y + 24\) was factored into \((8y-3)(3y-8)\).

Understanding polynomials allows us to break down complex expressions into simpler parts and solve equations more efficiently.

Fractions

Fractions are expressions representing the division of one quantity by another. They consist of a numerator and a denominator, separated by a slash. In algebra, fractions often involve variables and expressions rather than just numbers.

• Common Denominator: In the original problem, both fractions had the same denominator, \(8y - 3\). This is essential when adding or subtracting fractions, as it allows you to combine the numerators directly.
• Simplifying Fractions: Simplification is the process of canceling common factors from the numerator and denominator. In the exercise, the factor \(8y-3\) appeared in both, which could be canceled out.
• Operating on Fractions: Fractions can be added, subtracted, multiplied, or divided using specific rules. For subtraction, you'll often need a common denominator to simplify the operation.

Fractions, especially when involving algebraic terms, need careful manipulation to maintain their value while simplifying.

Rational Expressions

Rational expressions are similar to fractions but involve polynomials in the numerator, the denominator, or both. Simplifying rational expressions involves similar processes as simplifying numerical fractions.

• Simplification Process: To simplify a rational expression, factor both the numerator and the denominator (if possible) and cancel any common factors. In our solution, this meant factorizing the polynomial and canceling the \(8y-3\) from both parts of the fraction.
• Domain Considerations: Since rational expressions involve division, it's crucial to ensure that the denominator is not zero. For \(\frac{(8y-3)(3y-8)}{8y-3}\), the expression is undefined if \(8y-3=0\).
• Final Result Simplification: After cancelling common factors, you're left with the simplified result. In the example, it resulted in \(3y-8\).

Mastery of rational expressions involves practicing these simplifications and keeping an eye on potential restrictions.