Problem 29 - 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. $$ 8(x+y)^{3}(x-2 y), \quad 2(x-2 y) $$

Step-by-Step Solution

Answer

Question: Given the product of two factors, 8(x+y)^3 (x-2y), and one of the factors, 2(x-2y), find the other factor. Answer: 4(x+y)^3.

Step 1: Write the expression in the form "product = factor1 * factor2"

Since the first quantity given is the product, and the second quantity is one of the factors, we write the expression as follows: \(8(x+y)^3 (x-2y) = (unknown\ factor) * 2(x-2y)\).

Step 2: Divide both sides of the equation by the known factor to find the unknown factor

Now, we will divide both sides of the equation by the known factor, 2(x-2y), in order to solve for the unknown factor: \( \frac{8(x+y)^3 (x-2y)}{2(x-2y)} = unknown\ factor\).

Step 3: Simplify the expression to find the unknown factor

Now we will simplify the expression: \( \frac{8(x+y)^3 (x-2y)}{2(x-2y)} = 4(x+y)^3\). Thus, the unknown factor is: \(unknown\ factor = 4(x+y)^3\).

Key Concepts

Algebraic Expressions

In the realm of algebra, an algebraic expression is a combination of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, division, and exponentiation. Developing a deep understanding of these expressions is crucial because they form the foundation of algebra. For example, in the exercise provided, 8(x+y)^3(x-2y) is an algebraic expression that represents a quantity obtained by multiplying several factors together.

Algebraic expressions can often appear complicated, but by identifying the structure—coefficients, terms, and the combination of these through various operations—students can learn to manipulate and work with them effectively. It is important to recognize the parts of an expression to factorize it correctly, as we do when finding an unknown factor in the given product. A strong grasp of algebraic expressions allows students to solve a wide range of algebra problems with confidence.

Simplifying Expressions

Simplifying algebraic expressions is a fundamental skill in algebra. To simplify an expression means to make it as compact or as simple as possible, usually by performing arithmetic operations and combining like terms. In the context of the provided exercise, simplification was achieved by dividing the given algebraic expression by one of its factors.

This process effectively reduced the complexity of the expression, making it clearer that the answer was 4(x+y)^3. Simplifying an expression doesn't change its value—it just presents it in a different form. One can simplify expressions by

• cancelling out common factors
• combining like terms
• applying the distributive property

. Simplification can help in understanding the relationship between various parts of an expression and is an essential step toward problem solving in algebra.

Polynomial Factorization

Factorization is a key concept in algebra, especially when dealing with polynomial expressions. Polynomial factorization involves breaking down a complicated polynomial into simpler, irreducible factors. It is essentially the reverse process of expanding polynomials. For instance, in the provided exercise, we are asked to find an unknown factor of a polynomial product, which requires an understanding of how polynomials can be decomposed into factors.

Factorization techniques include:

• Factoring out the greatest common factor (GCF)
• Factoring by grouping
• Factoring trinomials
• Factoring by special products

Understanding how to employ these techniques is essential for solving higher-level algebra problems. In the problem, we performed a simple division to isolate the unknown factor, which is a form of factoring out a common factor. Mastering polynomial factorization equips students with tools to handle more complex algebraic problems and is pivotal in expanding their mathematical prowess.