Problem 49 - Elementary Algebra

Question

For the following problems, the first quantity represents the product and the second quantity a factor. Find the other factor. $$ 20 a^{2} b^{2}-10 a^{2}, \quad-10 a^{2} $$

Step-by-Step Solution

Answer

Answer: The other factor is $$-2 b^{2} + 1$$.

Step 1: Write down the given product and factor

We are given the following product and factor: $$ Product = 20 a^{2} b^{2} - 10 a^{2} $$ $$ Factor1 = -10 a^{2} $$ Now, we need to find the other factor (Factor2).

Step 2: Rearrange the equation to isolate Factor2

We have: $$ Product = Factor1 \times Factor2 $$ To find Factor2, we can rearrange the equation as follows: $$ Factor2 = \frac{Product}{Factor1} $$

Step 3: Substitute the given product and factor1 and compute Factor2

Now we substitute the given values into the equation: $$ Factor2 = \frac{20 a^{2} b^{2} - 10 a^{2}}{-10 a^{2}} $$ Let's simplify the equation by dividing both terms in the numerator by -10 a^{2}: $$ Factor2 = \frac{20 a^{2} b^{2}}{-10 a^{2}} - \frac{10 a^{2}}{-10 a^{2}} $$ Now we can simplify further: $$ Factor2 = -2 b^{2} + 1 $$ So the other factor is: $$ Factor2 = -2 b^{2} + 1 $$

Key Concepts

Algebraic Fractions

Algebraic fractions are similar to regular fractions, but instead of just numbers, they consist of algebraic expressions in the numerator and/or the denominator. To work with algebraic fractions, you apply the same rules as you do with numerical fractions, such as simplifying, adding, subtracting, multiplying, and dividing.
When dealing with algebraic fractions, the simplification process often involves factoring. Factoring is the process of rewriting an expression as a product of its factors.

• To simplify an algebraic fraction, you identify the common factors in both the numerator and the denominator.
• Once identified, you can cancel these common factors to reduce the fraction into its simplest form.
• It's important to note that you can only cancel factors; terms connected by addition or subtraction must first be factored.

Understanding how to handle algebraic fractions can make complex expressions more manageable and is a crucial skill in algebra.

Polynomial Division

Polynomial division is a method used to divide one polynomial by another, similar to long division with numbers. It's commonly used when simplifying algebraic expressions or solving algebraic equations. In this context, you can view polynomial division as the tool to find the missing factor when you're given a product and one known factor.
To perform polynomial division:

• Write down the dividend (the polynomial you're dividing) and the divisor (the polynomial you divide by).
• Divide the first term of the dividend by the first term of the divisor. This will give you the first term of the quotient.
• Multiply the entire divisor by the term obtained and subtract the result from the dividend.
• Repeat this process with the new polynomial until the degree of the remaining polynomial is less than the degree of the divisor.
• The quotient obtained is the result of the division.

Understanding these steps provides a foundation for dealing with more advanced algebraic problems and sets the stage for understanding algebraic fractions.

Simplifying Expressions

Simplifying expressions is a key component in solving algebraic problems, which involves reducing an expression to its simplest form. It often includes the processes of combining like terms and reducing expressions that have common factors.
The simplification process makes it easier to understand and solve algebraic equations. Here’s how you simplify expressions involving algebraic fractions:

• Identify and combine like terms. Like terms are terms whose variables (and their exponents) are the same.
• Look for common factors in both numerators and denominators, and factor them out.
• Cancel out these common factors to streamline the expression further.
• It's also useful to order the terms in a standard sequence, such as descending powers for polynomials.

By mastering the art of simplification, you can handle complex expressions more effectively and enhance your problem-solving skills.