Problem 62 - Algebra and Trigonometry

Question

31–76 ? Factor the expression completely. $$ 27 a^{3}+b^{6} $$

Step-by-Step Solution

Answer

The expression \(27a^3 + b^6\) factors to \((3a + b^2)(9a^2 - 3ab^2 + b^4)\).

Step 1: Identify Form of the Expression

The given expression is \(27a^3 + b^6\). Recognize that it is a sum of terms and try to identify if it fits any known pattern. It can be seen as a sum of cubes, \((3a)^3 + (b^2)^3\).

Step 2: Apply Sum of Cubes Formula

The sum of cubes formula is \(x^3 + y^3 = (x + y)(x^2 - xy + y^2)\). Here, identify \(x = 3a\) and \(y = b^2\). Apply the formula to split the expression.

Step 3: Substitute and Simplify

Substitute \(x = 3a\) and \(y = b^2\) into the sum of cubes formula: \((3a + b^2)((3a)^2 - (3a)(b^2) + (b^2)^2)\).

Step 4: Calculate Each Term in the Expression

Calculate each term in the expression:- The first term is \((3a + b^2)\).- The second term is \((3a)^2 = 9a^2\).- The third term is \((3a)(b^2) = 3ab^2\).- The fourth term is \((b^2)^2 = b^4\).

Step 5: Write the Final Factored Expression

Combine all the calculated terms: \((3a + b^2)(9a^2 - 3ab^2 + b^4)\). This is the fully factored form of the original expression.

Key Concepts

Sum of Cubes

In algebra, recognizing special patterns in polynomial expressions can simplify the factoring process. A common pattern is the "sum of cubes." This occurs when two terms, each raised to the power of three, are added together. The expression \((x^3 + y^3)\) is defined as a sum of cubes. To factor it, we use a specific formula:

• \(x^3 + y^3 = (x + y)(x^2 - xy + y^2)\)

Understanding this formula is essential. The first part, \((x + y)\), is straightforward, representing the sum of the cube roots. The second part, \((x^2 - xy + y^2)\), may seem more complex, but it is a quadratic expression. Each term in this expression arises naturally from expanding and rearranging the polynomial.When applying this formula to any cubic sum, substitute the terms that are perfect cubes into \(x\) and \(y\). This formula is a powerful tool that saves time and effort in simplifying cubic expressions.

Algebraic Expressions

Algebraic expressions represent a combination of numbers, variables, and operations. They can describe a wide range of mathematical phenomena, from simple arithmetic to complex polynomials. In the expression \(27a^3 + b^6\), variables \(a\) and \(b\) are tied to coefficients 27 and 1, respectively.Identifying algebraic properties such as degree, terms, and structure in expressions is crucial for understanding and factoring them:

• Degree: The greatest exponent in a single expression. Here, both terms have an equivalent degree when viewed as a sum of cubes.
• Terms: An expression's elements, separated by additions or subtractions. The terms involve constants raised to power, multiplied by variables.

When tackling problems involving algebraic expressions, recognizing these factors simplifies the understanding and factoring process.

Factoring Techniques

Factoring is breaking down an expression into simpler components, called factors, that when multiplied together return the original equation. Various techniques can be used depending on the structure of the polynomial.For sum of cubes like \(27a^3 + b^6\), specific steps are followed:

• Identify the pattern: Determine if it matches a known formula, like \((3a)^3 + (b^2)^3\).
• Substitute: Assign appropriate cubes to variables \(x\) and \(y\) and apply the formula \((x^3 + y^3 = (x + y)(x^2 - xy + y^2))\).
• Simplify: Expand and simplify using algebraic operations. In our problem, this results in \((3a + b^2)(9a^2 - 3ab^2 + b^4)\).

Each step ensures the expression reduces methodically to its simplest form. Using these techniques consistently will refine your skills in factoring polynomials effectively.