Problem 108 - College Algebra

Question

Factor completely. $$ (y+1)^{3}+1 $$

Step-by-Step Solution

Answer

\(y^3 + 3y^2 + 3y + 2 = (y+2)(y^2 + y + 1)\) is the factorized form of the original cubic trinomial.

Step 1: Identify a and b

In the given exercise, it can be observed that it represents the sum of two cubes. Comparing this with the standard form of the sum of two cubes, we get \(a = y+1\) and \(b = 1\).

Step 2: Apply the formula for the sum of cubes

The sum of two cubes can be factored using the formula \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\). Replacing the values of \(a\) and \(b\) in the formula, we get \((y+1+1)[(y+1)^2 - (y+1) + 1^2]\). This simplifies to \((y+2)[(y^2 + 2y + 1) - (y + 1) + 1]\).

Step 3: Simplify the expression

Simplify by combining like terms together. This gives \((y+2)(y^2 + y + 1)\). This is the factorized form of the original cubic trinomial

Key Concepts

Sum of Cubes

Factoring the sum of cubes involves identifying expressions of the form \(a^3 + b^3\). This is a special case in algebra where we can break down a cubic expression into simpler components. In our exercise, we have the expression \((y+1)^3 + 1^3\). Here, you can see this fits the form where \(a = y+1\) and \(b = 1\). The sum of cubes can always be factored using the formula:

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

This means we take our values \(a\) and \(b\) and apply them into this formula for effective factoring. With this method, it becomes much simpler to convert a cubic expression into a product of polynomials. Once factored, these expressions become more manageable and provide solutions in various algebraic problems.

Factoring Methods

Factoring is a critical method in algebra used to simplify expressions and solve equations. Different types of polynomials require different factoring methods. The formula applied in this exercise—\(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\)—is one such method specifically for cubics. This transforms complex expressions into more usable parts through

• Identifying parts \(a\) and \(b\)
• Substituting these into simple binomial and trinomial products

Once the values of \(a\) and \(b\) are determined, substitute these in:

• For \((a+b)\), you simply add \(a\) and \(b\)
• For \(a^2 - ab + b^2\), carefully compute the values with the identified \(a\) and \(b\)

Factoring using this method ensures that cubic expressions can be broken down and solved effectively, serving as a 'toolkit' approach in algebra.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations such as addition or multiplication. They form a major part of algebra and are the building blocks for more advanced mathematical problems.The given expression, \((y+1)^3 + 1\), begins as a cubic expression and through factoring, can be simplified into a more digestible format, offering clearer insight into its behavior and solution.

• These expressions can often be simplified or manipulated using established formulas like the sum of cubes.
• Simplification helps to reveal roots or solutions directly or to make further calculations easier.

Working with algebraic expressions, particularly through factoring, is a foundational skill that paves the way for more complex operations such as solving equations and inequalities.