Problem 108 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The completely factored form of the expression \((y+1)^3+1\) is \((y+2)(y^2+1)\).

Step 1: Identify the Cubes.

Let's recognize the given expression as a sum of cubes. Here, \(a^3\) can be identified as \((y+1)^3\) and \(b^3\) as \(1^3\), meaning \(a=y+1\) and \(b=1\).

Step 2: Apply the Sum of Cubes formula

The sum of cubes formula is \(a^3+b^3=(a+b)(a^2-ab+b^2)\). Plugging \(a=y+1\) and \(b=1\) into this formula, we get \((y+1+1)((y+1)^2-(y+1)*1+1^2)\).

Step 3: Simplify the Expression

Simplify each part of the obtained multiplication to get the final factored form. This translates to \((y+2)((y+1)^2-(y+1)+1)\). Further simplifying the expression within the second parentheses, this equals \((y+2)(y^2+1y+1-y+1)\). Combining like terms inside the parentheses leads to \((y+2)(y^2+1)\).

Key Concepts

Factoring

Factoring is a crucial skill in algebra that involves breaking down complex expressions into simpler parts, called factors, that can be multiplied together to get the original expression. It's like taking apart a puzzle to see what pieces make it up. One main goal of factoring is to make it easier to work with expressions when solving equations. In our exercise, we've focused on the sum of cubes, which is a specific type of factoring. The sum of cubes formula is \[a^3 + b^3 = (a + b)(a^2 - ab + b^2)\]This formula allows us to transform a seemingly complicated expression into a product of two simpler binomials and trinomials, making it much simpler to analyze or even solve. By mastering factoring techniques, you'll unlock the ability to handle a wide range of algebraic expressions with confidence.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that together represent a particular value or set of values. They can range from simple ones, like \(x+3\), to more complex forms, such as \[(y+1)^3 + 1\]which is the expression given in our exercise.
One of the powerful things about algebraic expressions is that they can be manipulated through operations like addition, subtraction, and factoring to reveal their underlying structure. This helps simplify expressions and solve equations. For instance, when dealing with the sum of cubes in our example, we rewrite the expression in a new form using the sum of cubes formula, which makes it easier to factor and further simplify. Recognizing the patterns and forms in algebraic expressions, such as cubes or differences, is a fundamental skill that will help in many areas of mathematics.

Polynomials

Polynomials are a key type of algebraic expression that involve sums and differences of terms, each consisting of a variable raised to a whole number exponent and a constant coefficient. They take the form \[a_nx^n + a_{n-1}x^{n-1} + \ ... + a_1x + a_0\]where each term is a monomial. Polynomials can vary in complexity from simple linear ones to high-degree forms.
In our exercise, the expression \[(y+1)^3 + 1\]was recognized and worked with as a polynomial, specifically a polynomial in a binomial form initially. Understanding and being able to manipulate polynomials using techniques like factoring are crucial in algebra as they appear frequently in equations, functions, and modeling real-world phenomena. Whether you're working to simplify expressions or solve equations, a solid grasp of polynomial structures and operations will make the process smoother and more intuitive.