Problem 26 - Algebra and Trigonometry

Question

\(21-28\) Use a Factoring Formula to factor the expression. $$ 1+1000 y^{3} $$

Step-by-Step Solution

Answer

The expression is factored as \((1 + 10y)(1 - 10y + 100y^2)\).

Step 1: Identify the Type of Expression

The expression given is similar to a sum of cubes. A sum of cubes has the general form \(a^3 + b^3\), and we need to factor this form.

Step 2: Rewrite the Expression in Sum of Cubes Form

Recognize \(1\) as \((1)^3\) and \(1000y^3\) as \((10y)^3\). So, the original expression \(1 + 1000y^3\) can be rewritten as \((1)^3 + (10y)^3\).

Step 3: Recall the Sum of Cubes Formula

The sum of cubes formula is given by: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). We will use this formula to factor the expression.

Step 4: Substitute into the Formula

Using \(a = 1\) and \(b = 10y\), substitute into the sum of cubes formula: \((1 + 10y)((1)^2 - (1)(10y) + (10y)^2)\).

Step 5: Simplify the Expression

Simplify the expression: \((1 + 10y)(1 - 10y + 100y^2)\). This is the fully factored form of the original expression.

Key Concepts

Sum of Cubes Formula

Understanding how to factor using the sum of cubes formula is a fundamental technique in algebra. The sum of cubes is an expression in the form of \(a^3 + b^3\). This expression can be factored using a specific formula:

• \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)

This formula helps break down the expression into simpler factors. To apply this formula, you first identify the values of \(a\) and \(b\) that make up the cubes in your original expression.

For instance, if you have \(1 + 1000y^3\), you recognize that \(1 = (1)^3\) and \(1000y^3 = (10y)^3\). With this recognition, you identify \(a = 1\) and \(b = 10y\). Substituting these values into the sum of cubes formula allows you to factor the expression successfully.

Algebraic Expressions

An algebraic expression consists of numbers, variables, and operation symbols. It's crucial to understand the structure of algebraic expressions to manipulate them correctly. These expressions can range from simple ones like \(x + 2\), to more complex ones like \(1 + 1000y^3\).

When dealing with an algebraic expression that resembles cubes, such as \(1 + 1000y^3\), you're looking for ways to simplify or rewrite it in a usable form—specifically, a factored form. Recognizing patterns in algebraic expressions, like that of cubes, makes it easier to use the appropriate formulas for simplification.

By rewriting the expression as \((1)^3 + (10y)^3\), you can apply the sum of cubes formula, which simplifies the tasks of finding manageable factors. The goal is to break down the complex expression into parts that are easier to handle.

Polynomial Factorization

Factoring polynomials is an essential skill that allows you to simplify expressions and solve equations more easily. For polynomial factorization, you often look for common patterns or use specific formulas, like the sum of cubes, to achieve your goal.

In our example, \(1 + 1000y^3\), the expression is recognized as a sum of cubes. Factoring this polynomial involves the steps of identifying the cubes, applying the sum of cubes formula, and simplifying the resulting expression. By rewriting it as \((1)^3 + (10y)^3\), and then substituting into the formula, you reveal the factored form: \((1 + 10y)(1 - 10y + 100y^2)\).

This process of breaking down the polynomial into simpler components is what makes polynomial factorization such a powerful tool in algebra. It not only simplifies the expressions but also aids in finding solutions to algebraic equations more effortlessly.