Problem 25 - Intermediate Algebra

Question

Perform each division. \(\frac{24 x^{6} y^{7}-12 x^{5} y^{12}+36 x y}{-48 x^{2} y^{3}}\)

Step-by-Step Solution

Answer

Simplified form: \(-\frac{1}{2}x^4y^4 + \frac{1}{4}x^3y^9 - \frac{3}{4}x^{-1}y^{-2}\).

Step 1: Factor the Numerator and Denominator

Start by factoring each term of the numerator, which is \(24x^6y^7 - 12x^5y^{12} + 36xy\). Notice a common factor of \(12xy\) for all terms. The numerator becomes \(12xy(2x^5y^6 - x^4y^{11} + 3)\). The denominator \(-48x^2y^3\) is already factored as a common term.

Step 2: Simplify the Fraction

Divide each factor in the numerator by the factors in the denominator. Specifically, divide \(12xy\) by \(-48x^2y^3\). This yields \(-\frac{1}{4}x^{-1}y^{-2}\). The expression now is: \(-\frac{1}{4}x^{-1}y^{-2}(2x^5y^6 - x^4y^{11} + 3)\).

Step 3: Distribute and Simplify Further

Multiply \(-\frac{1}{4}x^{-1}y^{-2}\) across the terms in parentheses: \[ -\frac{1}{4}x^{-1}y^{-2}(2x^5y^6) = -\frac{1}{2}x^4y^4 \]\[ -\frac{1}{4}x^{-1}y^{-2}(-x^4y^{11}) = \frac{1}{4}x^3y^9 \]\[ -\frac{1}{4}x^{-1}y^{-2}(3) = -\frac{3}{4}x^{-1}y^{-2} \]

Step 4: Combine the Results

Put together all the terms obtained in the previous step to write the simplified expression: \[-\frac{1}{2}x^4y^4 + \frac{1}{4}x^3y^9 - \frac{3}{4}x^{-1}y^{-2}\].

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial skill in algebra that involves breaking down a polynomial into simpler components or "factors." These factors, when multiplied together, give back the original polynomial. In the context of polynomial division, as seen in the original exercise, factoring the polynomials of the numerator and the denominator can greatly simplify the process. To factor effectively:

• Identify common factors in all the terms. In our exercise, notice the common factor 12xy in the numerator terms.
• Factor it out from the polynomial. The new expression will have fewer coefficients, making it easier to work with.

Factoring is particularly powerful because it reveals the structure of an expression. Once the structure is clear, simplification through division becomes less complex. So, whether you are dealing with numbers or variables, look for the greatest common divisor and factor it out confidently.

Simplifying Fractions

Simplifying fractions is the art of reducing fractions to their simplest form. This means canceling out common factors in the numerator and the denominator to make calculations easier and the formula more readable. In our exercise, once the polynomials are factored:

• Identify common factors in both the numerator and the denominator. Here, the factor 12xy in the numerator can be divided by the factor -48x²y³ in the denominator.
• Divide each term by these common factors, which changes the numeric coefficients and the exponents of the variables accordingly.

By simplifying in this manner, we were left with a coefficient of -1/4 and simplified exponents. This process not only clarifies the expressions but also significantly reduces the complexity of further operations. Remember, a simplified fraction is more efficient for both mathematical operations and communication!

Algebraic Expressions

Algebraic expressions involve a mix of numbers, variables, and operations. When tackling any algebra exercise, like polynomial division, understanding how to manipulate algebraic expressions is key. Look at what our exercise entailed:

• Beginning with a complex expression such as 24x⁶y⁷ - 12x⁵y¹² + 36xy, each term had to be adjusted based on the correspondingly factored terms.
• After simplifying, negative exponents like x^-1y^-2 are introduced, which can rearrange terms and give varying interpretations.

With algebraic expressions, clarity comes from understanding both the numeric coefficients and the arrangement of terms. Once you see how each part of an expression interacts or relates to others, working through algebraic problems can become both manageable and fulfilling. Always focus on the role of each variable and coefficient during simplification to ensure accuracy.