Problem 59
Question
Factor difference of cubes. \(x^{3}-216 y^{6}\)
Step-by-Step Solution
Verified Answer
The factorization is \((x - 6y^2)(x^2 + 6xy^2 + 36y^4)\).
1Step 1: Identify the Format
We recognize that we have a difference of cubes: \(x^3 - 216y^6\). The formula for factoring a difference of cubes is given by: \(a^3 - b^3 = (a-b)(a^2+ab+b^2)\).
2Step 2: Rewrite Each Term as a Cube
Recognize that \(x^3\) is already a perfect cube, with \(a = x\). For \(216y^6\), notice that \(216 = 6^3\) and \(y^6 = (y^2)^3\), so we can write \(b = 6y^2\).
3Step 3: Apply the Formula
Plugging in the values of \(a\) and \(b\) into the difference of cubes formula, we have: \(x^3 - (6y^2)^3 = (x - 6y^2)(x^2 + x(6y^2) + (6y^2)^2)\).
4Step 4: Simplify the Expression
Expand the second part of the factorization: the middle term becomes \(6xy^2\) (from \(x \cdot 6y^2\)), and the last term \((6y^2)^2\) simplifies to \(36y^4\). So, the factors simplify to \( (x - 6y^2)(x^2 + 6xy^2 + 36y^4)\).
Key Concepts
Understanding Algebraic ExpressionsExploring PolynomialsDiscovering Cube Roots
Understanding Algebraic Expressions
Algebraic expressions form the core of many mathematical problems, including those involving the factoring of the difference of cubes. At its most basic, an algebraic expression is a mathematical phrase that can contain numbers, variables (like \(x\) or \(y\)), and mathematical operators such as addition, subtraction, multiplication, and division. For example, the expression \(x^3 - 216y^6\) is composed of:
In essence, understanding algebraic expressions allows us to decipher complex problems incrementally, leading towards systematic and clear solutions.
- Two terms: \(x^3\) and \(-216y^6\),
- a subtraction operator between them.
In essence, understanding algebraic expressions allows us to decipher complex problems incrementally, leading towards systematic and clear solutions.
Exploring Polynomials
Polynomials are a special class of algebraic expressions where multiple terms can be connected by addition or subtraction. Each term in a polynomial is composed of a coefficient, a variable, and an exponent. For example, in the polynomial \(x^2 + 6xy^2 + 36y^4\), each term is as follows:
Polynomials are classified by their degree, which is the highest exponent of the variable(s) present in the polynomial. They are essential in algebra because they help form equations and functions used in various fields like physics, engineering, economics, and everyday problem-solving tasks. By factoring polynomials, we transform them into a simpler structure or their basic building blocks, which can help in resolving algebraic equations, optimizing functions, or even graphing them efficiently on a coordinate plane.
In this exercise, recognizing the form of polynomials as the difference of cubes is key to applying the correct factorization formula.
- \(x^2\) - the quadratic term in \(x\),
- \(6xy^2\) - the linear \(\text{xy}\) term with coefficient 6,
- \(36y^4\) - the quadratic term in \(y^4\).
Polynomials are classified by their degree, which is the highest exponent of the variable(s) present in the polynomial. They are essential in algebra because they help form equations and functions used in various fields like physics, engineering, economics, and everyday problem-solving tasks. By factoring polynomials, we transform them into a simpler structure or their basic building blocks, which can help in resolving algebraic equations, optimizing functions, or even graphing them efficiently on a coordinate plane.
In this exercise, recognizing the form of polynomials as the difference of cubes is key to applying the correct factorization formula.
Discovering Cube Roots
Cube roots are an extension of the square root concept applied to three dimensions. They provide a way to find a number which, when multiplied by itself twice more (i.e., cubed), equals the given number. For instance, the cube root of 8 is 2 because \(2 \times 2 \times 2 = 8\). In mathematical terms, the cube root of number \(n\) is written as \(\sqrt[3]{n}\).
In the exercise \(x^3 - 216y^6\), recognizing cube roots is vital for identifying what \(a\) and \(b\) represent in the difference of cubes. By realizing that \(x^3\) is the cube of \(x\) and \(216y^6\) is the cube of \(6y^2\), we easily adapt it to the formula \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\).
Since cubes are just products of the same factor three times, understanding cube roots assists in breaking down complex algebraic expressions into simpler factors. This insight is powerful, not only in algebra, but also in calculating volumes, dealing with exponential growth, and understanding natural phenomena that involve cubed quantities.
In the exercise \(x^3 - 216y^6\), recognizing cube roots is vital for identifying what \(a\) and \(b\) represent in the difference of cubes. By realizing that \(x^3\) is the cube of \(x\) and \(216y^6\) is the cube of \(6y^2\), we easily adapt it to the formula \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\).
Since cubes are just products of the same factor three times, understanding cube roots assists in breaking down complex algebraic expressions into simpler factors. This insight is powerful, not only in algebra, but also in calculating volumes, dealing with exponential growth, and understanding natural phenomena that involve cubed quantities.
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