Problem 58 - A Graphical Approach to College Algebra

Question

Factor each sum or difference of cubes completely. $$(b+3)^{3}-27$$

Step-by-Step Solution

Answer

The expression factors to \( b(b^{2} + 9b + 27) \).

Step 1: Identify the Sum or Difference of Cubes

The given expression is \( (b+3)^{3} - 27 \) which is a difference of cubes. We recognize this as \( a^{3} - b^{3} \)where \( a = b+3 \) and \( b = 3 \).

Step 2: Apply the Difference of Cubes Formula

The difference of cubes formula is:\( a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2}) \).Substitute \( a = b+3 \) and \( b = 3 \) into the formula:\( ((b+3) - 3)((b+3)^{2} + (b+3) imes 3 + 3^{2}) \).

Step 3: Simplify the First Factor

Simplify the expression \( (b+3) - 3 \) to \( b \).Thus, the first factor is \( b \).

Step 4: Expand and Simplify the Second Factor

Calculate the second factor: \((b+3)^{2} + (b+3) imes 3 + 3^{2}\).First, expand \((b+3)^{2} \) to get \(b^{2} + 6b + 9 \).Then, calculate \( (b+3) imes 3 = 3b + 9 \).Finally, \(3^{2} = 9\). Add these results together:\(b^{2} + 6b + 9 + 3b + 9 + 9 = b^{2} + 9b + 27 \).

Step 5: Combine the Factors

Combine the simplified first and second factors to get the factored form:\( b(b^{2} + 9b + 27) \).

Key Concepts

Difference of cubes

When confronted with expressions like \((b+3)^3 - 27\), we're dealing with a type called "difference of cubes." This refers to a scenario where you have something like \(a^3 - b^3\). It's crucial to unpack this thoroughly, because different formulas apply for sums and differences of cubes.
In our specific problem, we identify the expression as \(a^3 - b^3\) with \(a = b+3\) and \(b = 3\). Breaking down the expression helps us apply a specific algebraic formula for factorization. The difference of cubes formula is applied to simplify such expressions efficiently without expanding or simplifying everything manually. The formula is:

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

Memorizing this formula provides a fast track to solutions without extensive calculations. This understanding is pivotal in breaking complex cube expressions into simpler, solvable parts.

Polynomial expressions

Polynomial expressions consist of terms combined by addition or subtraction, involving variables raised to any power and multiplied by coefficients. In algebra, these expressions are the building blocks for more complex maths.
The expression \((b+3)\) in our problem is actually a polynomial because it can be expanded to a simpler polynomial \(b^2 + 9b + 27\). Understanding polynomials is fundamental because it lets us see how multiple terms relate to each other.
Expanding such polynomial expressions requires you to work with each part separately and then combine them to form larger expressions. It involves basic arithmetic operations on terms such as multiplication and addition/subtraction. Recognizing simple polynomial form helps us factor, simplifying algebraic processes and solving equations.

Algebraic formulas

Algebraic formulas serve as shortcuts in math that simplify processes like factoring or expanding. Without using such formulas, solving some mathematical expressions can be cumbersome.
The difference of cubes formula is one type of algebraic formula. It's used specifically for expressions like \(a^3 - b^3\), as seen in our example. By directly substituting the variables into the formula, we quickly get to the factored form without solving each component separately:

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

This process is invaluable because it streamlines hard-to-compute expressions into simpler ones. Learning these algebraic formulas provides powerful tools to handle a wide range of algebraic tasks, helping you see connections between numbers and variables faster and with greater ease.