Problem 106 - College Algebra with Modeling and Visualization

Question

Factor the expression completely. \(1+8 y^{3}\)

Step-by-Step Solution

Answer

The expression \(1 + 8y^3\) factors to \((1 + 2y)(4y^2 - 2y + 1)\).

Step 1: Identify the sum of cubes

The expression given is \(1 + 8y^3\). Notice that this expression can be expressed as the sum of cubes, since \(1 = 1^3\) and \(8y^3 = (2y)^3\). We then rewrite the expression as a sum of cubes: \(1^3 + (2y)^3\).

Step 2: Use the sum of cubes formula

The formula for factoring the sum of cubes \(a^3 + b^3\) is \((a + b)(a^2 - ab + b^2)\). In this case, assign \(a = 1\) and \(b = 2y\). Substitute into the formula: \((1 + 2y)((1)^2 - 1(2y) + (2y)^2).\)

Step 3: Simplify the expression

Now we simplify the expression using algebraic operations. Perform the calculations inside the compound factor:1. \((1)^2 = 1\)2. \(-1 \times 2y = -2y\)3. \((2y)^2 = 4y^2\)So the expression becomes:\[(1 + 2y)(1 - 2y + 4y^2).\]

Step 4: Combine the simplified terms

The factored form of the original expression \(1 + 8y^3\) is \((1 + 2y)(4y^2 - 2y + 1)\). Both parts of this product do not factor further, so this is the expression completely factored.

Key Concepts

Sum of Cubes

When dealing with algebraic expressions like the one in the exercise, identifying structure can simplify your task. The Sum of Cubes refers to expressions that can be written as the sum of two cube terms. In our case, the expression \(1 + 8y^3\) can be perceived as \(1^3 + (2y)^3\). Recognizing this pattern is key to factoring these types of expressions.

The formula for the sum of cubes, \(a^3 + b^3\), is formulated as:

• \((a + b)(a^2 - ab + b^2)\)

This formula might seem complex at first, but it breaks down the expression into a simplified, factored form. Understanding each component of this formula is critical, and once mastered, it offers a rather straightforward way to handle the sum of cubes. You identify \(a\) and \(b\) in your expression, plug them into the formula, simplify, and you're done!

Polynomial Expressions

Polynomial expressions are algebraic expressions consisting of terms involving variables raised to whole number powers and their coefficients. In the context of this exercise, our polynomial is \(1 + 8y^3\). Here, the powers are manageable, making it easier to apply factoring techniques.

Factoring helps to break down complex polynomial expressions into simpler factors, which can be quite handy in solving equations, simplifying expressions, or finding roots. Each term in a polynomial might not obviously show a factoring pattern unless you can see it as a product of simpler terms or as a well-known identity, like the sum of cubes.

• Identifying patterns or using identities often helps.
• Know common algebraic formulas to factor easily.

Overall, comfort with polynomials relies on practice and familiarity with algebraic rules.

Algebraic Formulas

Algebraic formulas are essential tools in simplifying and solving mathematical expressions. They serve as shortcuts to more complex operations, streamlining the process. In our task, we use the formula for the sum of cubes. Recognizing when and where to apply these formulas is integral to solving algebraic problems.

Typical formulas that you should memorize include:

• Sum of Cubes: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)
• Difference of Cubes: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)
• Perfect Square Trinomials, Quadratic Formulas, and more

Each formula applies to specific patterns or characteristics present in expressions. By practicing these, you'll gain efficiency in algebraic manipulations, preventing errors and saving time during tests or real-world problem-solving.