Problem 24 - Elementary and Intermediate Algebra

Question

Factor. $$ 27+y^{3} $$

Step-by-Step Solution

Answer

The factored form is \((3 + y)(9 - 3y + y^2)\).

Step 1: Identify the expression as a sum of cubes

The expression given is \(27 + y^3\). Observe that 27 is a perfect cube since it can be written as \(3^3\), and \(y^3\) is already a cube. Hence, we have a sum of cubes, which follows the standard form \(a^3 + b^3\). Here, \(a = 3\) and \(b = y\).

Step 2: Recall the formula for factoring a sum of cubes

The formula for factoring a sum of cubes \(a^3 + b^3\) is: \[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \] We will use this formula with the values \(a = 3\) and \(b = y\).

Step 3: Substitute into the sum of cubes formula

Using the formula: - The first factor is \(a + b\), which is \(3 + y\).- The second factor is \(a^2 - ab + b^2\): - \(a^2 = 3^2 = 9\) - \(ab = 3 \times y = 3y\) - \(b^2 = y^2\) - Thus, the second factor is \(9 - 3y + y^2\).

Step 4: Write the final factored form

Now write the expression using the factors obtained in step 3. The fully factored form of \(27 + y^3\) is:\[ (3 + y)(9 - 3y + y^2) \]

Key Concepts

Sum of Cubes

When we talk about the 'Sum of Cubes,' we are looking at expressions of the form \( a^3 + b^3 \). In algebra, recognizing these expressions can be immensely helpful because they have a specific factorization formula. Understanding the sum of cubes is crucial as it allows us to break down complex expressions into simpler factors, which can be easier to work with.

In our problem, the expression \( 27 + y^3 \) is a classic example of a sum of cubes. We identify this by noting that both \( 27 \) and \( y^3 \) are perfect cubes. The number 27 can be rewritten as \( 3^3 \), and \( y^3 \) is the cube of \( y \). Therefore, we see the structure fits perfectly with \( a^3 + b^3 \), where \( a = 3 \) and \( b = y \).

• The sum of cubes always follows the formula: \[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
• This formula helps to factor an otherwise inseparable polynomial expression.

Understanding the sum of cubes can simplify calculus, physics problems, and any algebra-related tasks that involve cubic expressions.

Factoring Formulas

Factoring formulas like those for the sum of cubes are powerful tools in algebra. They allow us to express complicated expressions as products of simpler factors. This is particularly helpful when solving equations or simplifying expressions.

In the given problem, we used the sum of cubes formula: \[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \] By substituting \( a = 3 \) and \( b = y \) from the problem, the expression \( 27 + y^3 \) can be broken down into its factors: \( (3 + y) \) and \( (9 - 3y + y^2) \).

Here is a quick breakdown of the calculation:

• First Factor: \( a + b = 3 + y \)
• Second Factor: Calculating step-by-step: - \( a^2 = 9 \) - \( ab = 3y \) - \( b^2 = y^2 \) Hence, Second Factor: \( 9 - 3y + y^2 \)

Factoring formulas like these are the building blocks of algebraic manipulation, allowing us to manage otherwise complex polynomial expressions.

Polynomial Expressions

Polynomial expressions are integral to understanding higher-level mathematics. They are expressions made up of terms, which include variables raised to a power and coefficients. When it comes to polynomials, factoring is a fundamental skill that helps simplify and solve equations.

In such expressions like \( 27 + y^3 \), identifying patterns such as a sum of cubes helps break down the polynomial into manageable parts. Doing so by recognizing its structure (cubes, quadratic forms, etc.) allows us to express the polynomial in factored form, making it easier to solve or simplify further.

With polynomial expressions:

• It's crucial to learn different factoring methods to simplify or solve equations.
• Recognizing forms like \( a^3 + b^3 \) is key to efficient factorization.

Polynomial expressions are everywhere in math, from simple algebra to complex calculus problems. Mastering factoring enhances our ability to manipulate these expressions effectively.