Problem 66 - Algebra and Trigonometry

Question

Factor the expression completely. $$ 27 a^{3}+b^{6} $$

Step-by-Step Solution

Answer

The complete factorization is \((3a + b^2)(9a^2 - 3ab^2 + b^4)\).

Step 1: Recognize the Sum of Cubes

The expression \( 27a^3 + b^6 \) can be recognized as a sum of cubes because \( 27a^3 = (3a)^3 \) and \( b^6 = (b^2)^3 \). This gives us the structure \( (3a)^3 + (b^2)^3 \).

Step 2: Apply the Sum of Cubes Formula

The sum of cubes formula is \( x^3 + y^3 = (x+y)(x^2 - xy + y^2) \). Here, \( x = 3a \) and \( y = b^2 \). Substitute these into the formula: \((3a + b^2)((3a)^2 - (3a)(b^2) + (b^2)^2)\).

Step 3: Simplify the Polynomial

Now, simplify the expression inside the parentheses:1. \((3a)^2 = 9a^2\)2. \(-(3a)(b^2) = -3ab^2\)3. \((b^2)^2 = b^4\)Substituting back, we get:\((3a + b^2)(9a^2 - 3ab^2 + b^4)\).

Step 4: Verify the Factorization

Double-check by expanding \((3a + b^2)(9a^2 - 3ab^2 + b^4)\) to ensure it simplifies back to \(27a^3 + b^6\). By distribution, the terms will expand and combine back to the original expression, confirming the factorization is correct.

Key Concepts

Sum of Cubes

The sum of cubes is a concept in algebra where two cubed terms are added together. Recognizing these sums allows us to simplify expressions by factoring them in a specific way. In our exercise, the components involved are \( 27a^3 \) and \( b^6 \), which can be rewritten as \( (3a)^3 \) and \( (b^2)^3 \) respectively. This aligns with the structure of the sum of cubes formula: \( x^3 + y^3 = (x+y)(x^2 - xy + y^2) \).
Applying this formula involves:

• Identifying \( x \) and \( y \) as bases of the cubed terms.
• Rewriting each term to show visibly it's a cube, such as \( (3a)^3 + (b^2)^3 \).
• Using the formula to break it into a factorable form.

By doing so, we can effectively factor expressions like \( 27a^3 + b^6 \) into simpler, more manageable parts.

Polynomial Algebra

Polynomial algebra is the area of mathematics that involves manipulating expressions with variables raised to whole-number exponents and combining them through operations like addition, subtraction, and multiplication.
In our current exercise, polynomial algebra is used to compose and simplify expressions like \( 27a^3 + b^6 \).

• Polynomials can include terms with variables, constants, and exponents.
• The expression in our example is a binomial—a polynomial with two terms, each a cube, which can be simplified using our factorization strategies.

Understanding polynomial algebra principles is key for restructuring expressions into more workable forms, aiding in both solving equations and interpreting algebraic functions.

Algebra Factorization

Algebra factorization is the process of breaking down a complex expression into simpler "factors" that multiply together to give back the original expression.In our exercise, we take the sum of cubes - \( 27a^3 + b^6 \) - and apply the sum of cubes formula to factor it. Factoring involves recognizing patterns and applying formulas to make a complex equation simpler.
Here's a breakdown:

• Identify the structure or special form of the polynomial, like the sum of cubes.
• Apply known algebraic identities or formulas, here it was \( x^3 + y^3 = (x+y)(x^2 - xy + y^2) \).
• Simplify the resultant factors, ensuring they multiply back to the original expression to verify correctness.

Mastering factorization opens up pathways to solve complex polynomial equations and forms the foundation for deeper algebra study.