Problem 22 - Algebra and Trigonometry

Question

17–24 ? Use a Factoring Formula to factor the expression. $$ 1+1000 y^{3} $$

Step-by-Step Solution

Answer

The expression \(1 + 1000y^3\) factors as \((1 + 10y)(1 - 10y + 100y^2)\).

Step 1: Recognize the Form

The expression \(1 + 1000y^3\) resembles the sum of cubes, which follows the formula \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). In this case, we can identify \(a = 1\) and \(b = 10y\) because \(1000 = (10)^3\).

Step 2: Substitute into the Sum of Cubes Formula

Now substitute \(a = 1\) and \(b = 10y\) into the sum of cubes formula: \(1 + 1000y^3 = (1 + 10y)((1)^2 - (1)(10y) + (10y)^2)\).

Step 3: Simplify the Terms

Multiply and simplify the terms inside the second factor: \((1 + 10y)(1 - 10y + 100y^2)\). This simplifies to \((1 + 10y)(1 - 10y + 100y^2)\), which is the factored form.

Key Concepts

Understanding the Sum of Cubes

The sum of cubes is a specific type of polynomial expression. It takes the form of \(a^3 + b^3\). This is one of several special polynomial identities used in algebra. The identity for the sum of cubes can be expressed as:

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

This formula is quite handy because it allows us to rewrite a sum of cubes as a product of two simpler polynomials. In the given exercise situation, \(1 + 1000y^3\) has been identified to match this pattern. Here, the terms \(1\) and \(1000y^3\) are each perfect cubes. This lets us align them with \(a\) and \(b\) as \(a = 1\) and \(b = 10y\), since \(1000 = (10)^3\). This identification is the key step in using this elegant factoring method.

Factored Form Explained

Factored form is an expression broken down into a product of its simplest components or factors. The advantage of working in the factored form is that it can make further calculations simpler, such as solving equations or simplifying expressions. In the given example, the factored form of \(1 + 1000y^3\) is derived by applying the sum of cubes formula. We started with:

• Original: \(1 + 1000y^3\)
• As sum of cubes: \((1) + (10y)^3\)
• Using the formula: \((1 + 10y)(1^2 - 1 \times 10y + (10y)^2)\)

The result of this calculation simplification is \((1 + 10y)(1 - 10y + 100y^2)\). This not only gives us a succinct expression in a structurally simplified form but also helps us see the roots of the polynomial and other algebraic properties tied to it.

Algebra and its Techniques

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating those symbols. Algebraic techniques are pivotal for solving equations, working with formulas, and performing various transformations like factoring. Factoring is incredibly useful in algebra because it transforms complex polynomial expressions into simpler ones that are easier to handle. Understanding how to recognize different forms, such as the sum of cubes, is crucial. It involves a mixture of pattern recognition and substitution, which are key skills in algebra. These skills are not only critical for high school math but are also foundational for more advanced studies and practical applications. By learning how to factor polynomials using these methods, students can enhance their problem-solving toolkit significantly. This knowledge becomes a stepping-stone to deeper algebraic topics such as quadratic equations, polynomial long division, and beyond.