Problem 31 - Elementary Algebra

Question

In the following problems, the first quantity represents the product and the second quantity represents a factor of that product. Find the other factor. $$ 26(x-5 y)^{10}(x-3 y)^{12}, \quad-2(x-5 y)^{7}(x-3 y)^{7} $$

Step-by-Step Solution

Answer

Answer: The second factor is -13(x-5y)^{3}(x-3y)^{5}.

Step 1: Identify the given product and factor

The given product is 26(x-5y)^{10}(x-3y)^{12}, and the known factor is -2(x-5y)^{7}(x-3y)^{7}.

Step 2: Divide the product by the given factor

Find the second factor by dividing the given product by the given factor: $$ \frac{26(x-5y)^{10}(x-3y)^{12}}{-2(x-5y)^{7}(x-3y)^{7}} $$

Step 3: Simplify the coefficients

Divide the constant terms 26 and -2: $$ \frac{26}{-2} = -13 $$

Step 4: Simplify the first expression (x-5y) terms

Use the properties of exponents to simplify the (x-5y) terms: $$ \frac{(x-5y)^{10}}{(x-5y)^{7}}= (x-5y)^{10-7}=(x-5y)^{3} $$

Step 5: Simplify the second expression (x-3y) terms

Use the properties of exponents to simplify the (x-3y) terms: $$ \frac{(x-3y)^{12}}{(x-3y)^{7}}= (x-3y)^{12-7}=(x-3y)^{5} $$

Step 6: Combine the simplified terms to find the other factor

Now that we have simplified all the components of the expression, we combine them to find the second factor: $$ -13(x-5y)^{3}(x-3y)^{5} $$ So, the other factor is \(-13(x-5y)^{3}(x-3y)^{5}\).

Key Concepts

Properties of Exponents

When working with expressions involving powers, the properties of exponents come in handy to simplify calculations. Exponents indicate how many times a number, known as the base, is multiplied by itself. Let's explore a couple of key exponent properties that are useful in solving algebraic problems.

1. **Product of Powers Property**: When you multiply two powers with the same base, you add their exponents. For example, in the case of i.e., \((a^m \cdot a^n = a^{m+n})\)

2. **Quotient of Powers Property**: When dividing two powers with the same base, you subtract the exponent of the denominator from the exponent of the numerator, i.e., \((\frac{a^m}{a^n} = a^{m-n})\).

In our exercise, we used the quotient of powers property to simplify \((x-5y)^{10}\) divided by \((x-5y)^{7}\) and \((x-3y)^{12}\) divided by \((x-3y)^{7}.\) By subtracting the exponents, these terms became \((x-5y)^3\) and \((x-3y)^5\) respectively. Understanding these properties makes manipulating expressions with exponents much easier and more efficient!

Polynomial Division

Polynomial division is similar to numerical division but involves expressions with variables. It helps to find factors or reduce complex expressions. Here's how we can approach polynomial division:

- **Division of Coefficients**: Begin by dividing the coefficients, just like you do with numbers. This involves dividing the leading terms in the expressions. For instance, in our example, \((26 \div -2 = -13)\). This gives us the leading coefficient of the resulting expression.

- **Apply Exponent Rules**: While dividing the variables (terms with exponents), apply the properties of exponents to simplify each component, which we discussed previously.

Polynomial division is key to efficiently simplifying algebraic expressions. By performing polynomial division properly, complex expressions can be broken down into simpler parts, making them more manageable for further calculation or analysis.

Factorization

Factorization is breaking down a complex expression into simpler, multiplyable parts, known as factors. Essentially, it is the reverse of expansion in algebra. It is crucial in simplifying expressions and solving equations.

- **Recognizing Common Factors**: Look for terms that can be factored out from an expression. In our exercise, \(-2(x-5y)^7(x-3y)^7\) was a common factor. Recognizing this allowed us to simplify the problem efficiently.

- **Use of Simplified Components**: Once you have identified common factors, divide the expression by these to find the remaining factors. As shown, after simplifying by dividing the product by the known factor, the other factor was \(-13(x-5y)^3(x-3y)^5\).

By understanding factorization, one can deconstruct polynomials into factors, making equations simpler and easier to solve. This method not only simplifies calculations but also provides insights into the properties that govern algebraic expressions.