Problem 77 - Beginning and Intermediate Algebra

Question

Factor completely. $$d^{3}+1$$

Step-by-Step Solution

Answer

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

Step 1: Identify the sum of cubes

In the given expression, we can see that it is in the form of sum of two cubes as \(a^3 + b^3\), where \(a^3 = d^3\) and \(b^3 = 1^3\). So, \(a = d\) and \(b = 1\).

Step 2: Apply the sum of cubes formula

Our given expression is of the form: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). Substitute a and b with the values we found in the previous step: \(d^3 + 1^3 = (d + 1)(d^2 - d(1) + 1^2)\)

Step 3: Simplify the expression

Now, let's simplify the expression: \(d^3 + 1 = (d + 1)(d^2 - d + 1)\) So, the completely factored form of the given expression \(d^3 + 1\) is \((d + 1)(d^2 - d + 1)\).

Key Concepts

Sum of Cubes

In mathematics, the sum of cubes is a powerful concept when it comes to factoring certain types of polynomial expressions. When you encounter a sum of cubes, it means you have two cube terms that are being added together, formatted as \(a^3 + b^3\). These types of expressions can be neatly factored using a specific formula.

The sum of cubes formula is:\[a^3 + b^3 = (a + b)(a^2 - ab + b^2)\]

To apply this formula, you must first recognize the expression as a sum of cubes. For instance, if you have \(d^3 + 1\), this can be rewritten as \(d^3 + 1^3\).

• Identify \(a\) and \(b\): Here, \(a = d\) and \(b = 1\).
• Substitute these values into the sum of cubes formula.

With the expression \(d^3 + 1\), substituting \(a\) and \(b\) gives you \((d + 1)(d^2 - d(1) + 1^2)\).

This way, you can simplify and factor the polynomial completely.

Factoring Techniques

Factoring techniques are essential tools when simplifying polynomial expressions. These methods enable you to break down complex equations into simpler parts that are easier to manage and solve. One such technique is the sum of cubes method, used for expressions like \(a^3 + b^3\).

Here are some general steps in factoring any polynomial expression:

• Look for a common factor among all terms. If there is one, factor it out.
• Identify the pattern (such as sum/difference of cubes or squares).
• Use the appropriate factoring formula for the identified pattern.
• Simplify the expression by performing the multiplication or other arithmetic needed.

For the expression \(d^3 + 1\), no common factor exists, but it fits the sum of cubes pattern. Therefore, using the formula \((a + b)(a^2 - ab + b^2)\) is the correct approach.

Choosing the right technique depends on your recognition of the expression's form and the formulas you need to apply.

Polynomial Expressions

Polynomial expressions are algebraic expressions that involve variables raised to whole number exponents. These expressions can be monomials, like \(3x^2\), or polynomials, like \(d^3 + 1\).

Polynomials can have multiple terms connected by addition or subtraction. When working with polynomials, you often need to simplify or factor them to make calculations easier. Polynomials have specific structures:

• Monomial: A single term (e.g., \(5x^2\)).
• Binomial: Two terms (e.g., \(x^2 + 3\)).
• Trinomial: Three terms (e.g., \(x^2 + 2x + 3\)).

In the case of \(d^3 + 1\), this is a binomial, as it consists of two terms. Using known methods like the sum of cubes is how you can factor such expressions.

Recognizing these structures helps you decide how to manipulate and break down the expressions, making your work with polynomials more manageable.