Problem 37 - Elementary Algebra

Question

For the following problems, the first quantity represents the product and the second quantity a factor. Find the other factor. $$ 9 a^{5}+6 a^{5}-18 a^{4}+24 a^{2}, \quad 3 a^{2} $$

Step-by-Step Solution

Answer

Question: Find the other factor given the product \(9 a^{5}+6 a^{5}-18 a^{4}+24 a^{2}\) and one of the factors \(3 a^{2}\). Answer: The other factor is \(5a^3 - 6a^2 + 8\).

Step 1: Write down the problem

We need to find the unknown factor (\(x\)) in the equation: $$ (3a^2) \times x = 9a^5 + 6a^5 - 18a^4 + 24a^2 $$

Step 2: Divide by the known factor

Divide the given product by the known factor (\(3a^2\)): $$ x = \frac{9a^5 + 6a^5 - 18a^4 + 24a^2}{3a^2} $$

Step 3: Distribute the division across all terms

Divide each term in the numerator by the common term \(3a^2\): $$ x = 3a^3 + 2a^3 - 6a^2 + 8 $$

Step 4: Simplify the expression

Combine like terms in the expression: $$ x = 5a^3 - 6a^2 + 8 $$

Step 5: Write down the final answer

The other factor is: $$ x = 5a^3 - 6a^2 + 8 $$

Key Concepts

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and operations like addition, subtraction, multiplication, and division. In essence, they are mathematical phrases that can include constants (like the number 3 in our problem) and coefficients (numbers multiplying variables). For instance, in the expression \( 9a^5 + 6a^5 - 18a^4 + 24a^2 \), each term involves a coefficient and a variable part raised to a power.

• The variable part, like \(a^5\), shows the variable \(a\) raised to the fifth power.
• The coefficient, such as 9, is what multiplies the variable part.

When working with algebraic expressions, you often simplify them by combining like terms. Like terms are terms that have the same variable raised to the same power. This means you can easily add or subtract their coefficients.
Understanding how to manipulate algebraic expressions is crucial in solving polynomial equations, which we will explore next.

Division of Polynomials

The division of polynomials involves splitting a complex polynomial expression by another polynomial, much like regular division but with more steps. In this context, a polynomial is an algebraic expression consisting of multiple terms. To divide a polynomial by another polynomial like \(3a^2\), follow these simple steps:

• Write the division as a fraction: \( \frac{9a^5 + 6a^5 - 18a^4 + 24a^2}{3a^2} \).
• Separate each term in the numerator, dividing it by the term in the denominator.
• Simplify each resulting term individually.

This method is similar to distributing the denominator to each term in the numerator, simplifying the expression step-by-step. It's essential to divide each term correctly, accounting for both the coefficients and the variable parts. When dividing terms like \(9a^5 \div 3a^2\), you're basically simplifying the coefficients (9 divided by 3, which is 3) and subtracting the powers of \(a\) (\(a^5 - a^2 = a^3\)). This careful process eventually leads you to the simplified expression.

Simplifying Expressions

Simplifying expressions is about making them as straightforward as possible, often by combining like terms or removing unnecessary complexity. Once you've divided a polynomial expression, the next step is to combine any like terms within the expression. In the example given, after division, you obtain \( x = 3a^3 + 2a^3 - 6a^2 + 8 \).

• Identify like terms: In this case, \(3a^3\) and \(2a^3\) are like terms because they share the same variable raised to the same power.
• Combine these terms by adding their coefficients: \(3a^3 + 2a^3 = 5a^3\).
• Check for other opportunities to simplify: The result is \(5a^3 - 6a^2 + 8\), where no further simplification is possible.

Simplification makes the expression easier to work with and helps reveal any inherent relationships or solutions within the mathematical problem. This process is vital in algebra, enabling you to find more natural solutions and insights.