Problem 80 - Algebra and Trigonometry with Analytic Geometry

Question

Factor the polynomial. $$ 216 x^{9}+125 y^{3} $$

Step-by-Step Solution

Answer

The polynomial factors as \((6x^3 + 5y)(36x^6 - 30x^3y + 25y^2)\).

Step 1: Identify the Polynomial Type

The given polynomial is \( 216x^9 + 125y^3 \). Notice this is a sum of two cubes, where \( 216x^9 = (6x^3)^3 \) and \( 125y^3 = (5y)^3 \). So, we recognize it as a sum of cubes: \( (a^3 + b^3) \), where \( a = 6x^3 \) and \( b = 5y \).

Step 2: Apply Sum of Cubes Formula

The formula for the sum of cubes is \( a^3 + b^3 = (a+b)(a^2 - ab + b^2) \). Using \( a = 6x^3 \) and \( b = 5y \), we substitute these values into the formula.

Step 3: Perform Substitution and Simplification

Substitute \( a = 6x^3 \) and \( b = 5y \) into the sum of cubes formula:- First term: \( a+b = 6x^3 + 5y \)- Second term: \( a^2 - ab + b^2 = (6x^3)^2 - (6x^3)(5y) + (5y)^2 \) Simplify the expressions:- \( (6x^3)^2 = 36x^6 \)- \( (6x^3)(5y) = 30x^3y \)- \( (5y)^2 = 25y^2 \) That results in \( a^2 - ab + b^2 = 36x^6 - 30x^3y + 25y^2 \).

Step 4: Write the Final Factored Form

Combining the results from the above steps using the formula, the factored form of \( 216x^9 + 125y^3 \) is:\[ (6x^3 + 5y)(36x^6 - 30x^3y + 25y^2) \].

Key Concepts

Sum of Cubes Formula

The "Sum of Cubes Formula" is an essential tool in algebra for factoring expressions where two cubic terms are added together. This formula helps to break down complex algebraic expressions into simpler parts. The formula is written as:

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

Using this formula makes it possible to rewrite cubic expressions, facilitating easier manipulation either for solving equations or simplifying expressions. By identifying each cube in the expression, we label them as \( a \) and \( b \). For example, in the expression \( 216x^9 + 125y^3 \), we identify \( 216x^9 \) as \( (6x^3)^3 \) and \( 125y^3 \) as \( (5y)^3 \). The formula provides a structured way to address problems involving cubes.

Algebraic Expressions

Algebraic expressions are the building blocks of algebra, consisting of numbers, variables, and arithmetic operations. They can range from simple, like \( x + 2 \), to complex, like \( 216x^9 + 125y^3 \). Working with these expressions involves a few different skills:

• Identifying terms and their components
• Using arithmetic operations to simplify
• Recognizing patterns like cubes or squares

In the context of polynomial factoring, algebraic expressions are manipulated using formulas like the sum of cubes. This unique factorization splits the expression into components that are easier to handle and solve. Understanding how to interpret and reconfigure these expressions is crucial in algebra.

Polynomial Identities

Polynomial identities are equations that are true for all values of the variables. Recognizing these identities helps in simplifying and factoring polynomials effectively. Important identities include:

• Sum of cubes: \( a^3 + b^3 = (a+b)(a^2 - ab + b^2) \)
• Differences of squares: \( a^2 - b^2 = (a-b)(a+b) \)

These identities offer a foundation for putting expressions into simpler forms during algebraic manipulation. For instance, when using the sum of cubes, you rely on knowing that any expression fitting that pattern can be reduced according to the identity. This turns potentially convoluted expressions into organized, manageable ones, making problem-solving more straightforward.

Cubic Polynomials

Cubic polynomials are expressions where the highest power of the variable is three, like \( ax^3+bx^2+cx+d \). In algebra, these polynomials can express any number of practical scenarios and require specific techniques to factor or simplify:

• Identifying if terms can be grouped or factored
• Recognizing patterns such as the sum or difference of cubes
• Using known formulas and identities to simplify

Factoring cubic polynomials involves examining each term to determine their base components, such as \( (6x^3)^3 \), to apply suitable algebraic identities. Successfully working with cubic polynomials encourages not just rote memorization of formulas, but also a deeper recognition of underlying structures within algebraic expressions.