Problem 39 - Elementary Algebra

Question

For the following problems, the first quantity represents the product and the second quantity a factor. Find the other factor. $$ 12 x^{3} y^{5}+20 x^{3} y^{2}, \quad 4 x^{3} y^{2} $$

Step-by-Step Solution

Answer

Answer: The other factor is $$3y^{3} + 5$$.

Step 1: Understand the problem

We need to find the other factor by dividing the product with the given factor. The product is: $$ 12x^{3}y^{5} + 20x^{3}y^{2} $$ and the given factor is: $$ 4x^{3}y^{2} $$

Step 2: Divide each term of the product by the given factor

To find the other factor, we need to divide each term of the product by the given factor. This can be done by dividing the coefficients and the variables for each term. $$ \frac{12x^{3}y^{5}}{4x^{3}y^{2}} + \frac{20x^{3}y^{2}}{4x^{3}y^{2}} $$

Step 3: Simplify the division of coefficients and variables

Now let's simplify the division for coefficients and variables in each term: $$ \frac{12}{4} \cdot \frac{x^{3}}{x^{3}} \cdot \frac{y^{5}}{y^{2}} + \frac{20}{4} \cdot \frac{x^{3}}{x^{3}} \cdot \frac{y^{2}}{y^{2}} $$

Step 4: Calculate and simplify the results for each term

Next, we will perform the calculations for each term and simplify the results: $$ 3 \cdot 1 \cdot y^{(5-2)} + 5 \cdot 1 \cdot 1 $$

Step 5: Write the final answer

Finally, let's write the answer as the other factor: $$ 3y^{3} + 5 $$ So, the other factor of the given product is: $$ 3y^{3} + 5 $$

Key Concepts

Algebraic Division

Algebraic division is a fundamental process in algebra that involves dividing one algebraic expression by another. This procedure is akin to regular numerical division but involves variables. The goal is to find the quotient, or result, when one polynomial or algebraic expression is divided by another.

• To accomplish this, each term in the dividend (the expression being divided) undergoes division by the divisor (the given factor).
• The process focuses on dividing both the numerical coefficients and variable parts.
• Simplification is key, as it provides clarity and exposes the underlying algebraic structure.

In our example of algebraic division with the polynomial \(12x^3y^5 + 20x^3y^2\), the given factor is \(4x^3y^2\). Each term of the polynomial is handled separately. This involves cleaning up coefficients and simplifying variables by reducing powers, just as demonstrated in the step-by-step solution. This results in the clean quotient \(3y^3 + 5\). This process highlights how various algebraic elements interact and can be restructured.

Polynomials

Polynomials are expressions that consist of variables, coefficients, and constants that are combined using addition, subtraction, and multiplication. Each part of a polynomial is called a term, and these terms are composed of:

• Coefficients: The numerical factor associated with each term.
• Variables: The letters (like \(x\) and \(y\)) that can change values.
• Exponents: Numbers indicating the power to which the variable is raised.

Polynomials can be simple, like \(x + 2\), or more complex, involving multiple terms and variables like \(12x^3y^5 + 20x^3y^2\). In our example, the polynomial includes terms with similar variables, \(x^3y^5\) and \(x^3y^2\), each having their unique combination of coefficients and variables. Understanding polynomials is essential knowing how to manipulate them, such as through addition, subtraction, and division, as practiced in exercises like this one.

Factorization

Factorization is the process of breaking down an expression into a product of simpler expressions or factors. This is a vital skill in algebra that simplifies solving equations and understanding polynomial structures.

• Finding common factors is the first step. In algebra, factors could be numerical or involve variables.
• For any polynomial, factorization involves expressing it as a product of its factors.
• Simplifying complicated expressions reveals patterns and potential solutions.

In the exercise, we factorized the polynomial \(12x^3y^5 + 20x^3y^2\) by dividing it by the factor \(4x^3y^2\) to find the other factor, which is \(3y^3 + 5\). This demonstrates how factorization can reduce complex algebraic forms to more recognizable and manageable pieces. It is especially useful when solving equations or simplifying expressions for other operations.