Problem 72 - Beginning and Intermediate Algebra

Question

Factor completely. $$32 n^{5}-200 n^{3}$$

Step-by-Step Solution

Answer

The completely factored expression is: \(32n^{5} - 200n^{3} = 8n^{3}(2n + 5)(2n - 5)\)

Step 1: Factor out the greatest common divisor (GCD) of the coefficients

We first need to determine the GCD of the coefficients 32 and 200. The GCD of 32 and 200 is 8. So, we can factor out the GCD from the given expression: $$32n^{5} - 200n^{3} = 8(4n^{5} - 25n^{3})$$

Step 2: Factor out the common variable term

We can now factor out the common variable term, which is \(n^{3}\), from the expression inside the parenthesis: $$8(4n^{5} - 25n^{3}) = 8n^{3}(4n^{2} - 25)$$

Step 3: Identify the difference of squares

Notice that the expression inside the parenthesis is a difference of squares, where \(4n^{2}\) represents \((2n)^2\) and \(25\) represents \((5)^2\). We can apply the difference of squares formula, which states that \(a^2 - b^2 = (a+b)(a-b)\), to this term: $$8n^{3}(4n^{2} - 25) = 8n^{3}[(2n + 5)(2n - 5)]$$ The given expression has been factored completely as: $$32n^{5} - 200n^{3} = 8n^{3}(2n + 5)(2n - 5)$$

Key Concepts

Greatest Common Divisor

In mathematics, the greatest common divisor (GCD), also known as the greatest common factor (GCF), is the largest positive integer that divides two or more integers without leaving a remainder. To find the GCD of two numbers, such as 32 and 200, we must identify the largest number that divides both completely.

• For 32, the divisors are: 1, 2, 4, 8, 16, 32.
• For 200, the divisors are: 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 200.

In this case, the greatest number that appears in both lists is 8. Therefore, 8 is the GCD of 32 and 200. When factoring polynomials, identifying the GCD of the coefficients is the first step. This allows us to simplify the expression by pulling out the common factor, making the polynomial simpler and easier to work with.

Difference of Squares

The difference of squares is a unique type of binomial that can be factored in a special way. It occurs when you have two perfect squares subtracted from each other. The formula for factoring a difference of squares is:\[a^2 - b^2 = (a+b)(a-b)\]This property is useful when dealing with expressions such as \(4n^2 - 25\). Here,

• \(4n^2\) is the same as \((2n)^2\)
• \(25\) is the same as \((5)^2\)

Recognizing this allows you to rewrite the expression as \((2n + 5)(2n - 5)\), making it easier to factor the polynomial completely.
This technique is a powerful tool in algebraic manipulation and helps transform complicated expressions into more manageable parts.

Algebraic Expressions

Algebraic expressions are mathematical phrases that include numbers, variables, and operations such as addition, subtraction, multiplication, and division. They form the backbone of algebra and are essential for solving equations and modeling real-world problems.When working with expressions like \(32n^5 - 200n^3\), it's important to understand each component:

• The coefficients (32 and 200) denote numerical factors of the variable terms.
• Variables and their exponents indicate the degree of each term. In this case, \(n^5\) and \(n^3\) tell us the power to which the variable is raised.

By combining these components, we can manipulate and simplify expressions. Factoring is a key method used to make expressions simpler or to solve equations, often transforming a complex algebraic expression into a product of simpler factors. Understanding how to work with and manipulate these expressions is crucial for success in algebra.