Problem 30 - 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. $$ 14(a-3)^{6}(a+4)^{2}, \quad 2(a-3)^{2}(a+4) $$

Step-by-Step Solution

Answer

Answer: \(7(a-3)^{4}(a+4)\)

Step 1: Rewrite the problem

Begin by rewriting the problem as follows: Given the product \(14(a-3)^{6}(a+4)^{2}\) and one factor \(2(a-3)^{2}(a+4)\), find the other factor.

Step 2: Set up the equation

We know that the product is equal to the multiplication of both factors. So, we can set up an equation to solve for the other factor: $$ 14(a-3)^{6}(a+4)^{2} = 2(a-3)^{2}(a+4)(\text{Other Factor}) $$

Step 3: Divide both sides by the given factor

Now we can find the other factor by dividing both sides of the equation by the given factor: $$ \frac{14(a-3)^{6}(a+4)^{2}}{2(a-3)^{2}(a+4)} = \text{Other Factor} $$

Step 4: Simplify the expression

Simplify the expression on the left side of the equation using the cancellation properties and the rules of exponents: $$ \frac{14(a-3)^{6}(a+4)^{2}}{2(a-3)^{2}(a+4)} = 7(a-3)^{4}(a+4) $$ The left side is now simplified.

Step 5: Write the answer

As the left side of the equation is now simplified, we can write the other factor as: $$ \text{Other Factor} = 7(a-3)^{4}(a+4) $$ So, the other factor is \(7(a-3)^{4}(a+4)\).

Key Concepts

Exponents

Exponents are a fundamental part of algebra that represent repeated multiplication. In the problem, expressions like \((a-3)^6\) and \((a+4)^2\) demonstrate how they work. Here:

• \((a-3)^6\) means \((a-3)\) multiplied by itself six times.
• \((a+4)^2\) means \((a+4)\) multiplied by itself twice.

Using exponents simplifies expressions and helps calculations. The rules of exponents allow us to manipulate these numbers easily. For example:

• When dividing exponents with the same base, subtract: \(a^m / a^n = a^{m-n}\).
• When multiplying exponents with the same base, add: \(a^m \times a^n = a^{m+n}\).

In this exercise, we simplify \((a-3)^6 / (a-3)^2\) by subtracting the exponents to get \((a-3)^4\). This powerful tool makes handling large expressions more manageable and helps solve complex problems efficiently.

Algebraic Simplification

Algebraic simplification involves reducing expressions to their simplest form. It's done by canceling common factors and applying mathematical rules.In the given exercise:

• The expression \(\frac{14(a-3)^6(a+4)^2}{2(a-3)^2(a+4)}\) needs to be simplified to find the other factor.
• First, \(14/2\) simplifies to \(7\).
• Then, cancel out the common terms: \((a-3)^6 / (a-3)^2 = (a-3)^4\) and \((a+4)^2/(a+4) = (a+4)\).

This results in the expression \(7(a-3)^4(a+4)\), which is much simpler. Simplification reduces complexity and keeps equations easier to work with. It's all about finding patterns and applying consistent rules.

Polynomial Division

Polynomial division is similar to dividing numbers. Here, you're dividing one polynomial by another to find remaining factors. This exercise gives:

• The product: \(14(a-3)^6(a+4)^2\)
• A given factor: \(2(a-3)^2(a+4)\)

We want to find the other factor by dividing the product by the given factor.Substituting these into the division, we performed:

• Divide coefficients: \(14/2 = 7\).
• Simplify exponents \((a-3)^6/(a-3)^2 = (a-3)^4\) and \((a+4)^2/(a+4) = (a+4)\).

The result \(7(a-3)^4(a+4)\) is the other factor left after division.It's like dividing a cake to see how many slices are left after giving some away. Understanding polynomial division helps in factorizing and solving equations efficiently.