Problem 82 - Algebra and Trigonometry with Analytic Geometry

Question

Factor the polynomial. $$ x^{6}-27 y^{3} $$

Step-by-Step Solution

Answer

The polynomial \( x^6 - 27y^3 \) factors to \( (x^2 - 3y)(x^4 + 3x^2y + 9y^2) \).

Step 1: Recognize the Form

Notice that the polynomial \( x^6 - 27y^3 \) can be expressed as a difference of two cubes. It matches the pattern \( a^3 - b^3 \) where \( a = x^2 \) and \( b = 3y \).

Step 2: Recall the Factoring Formula

The difference of cubes can be factored using the formula \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \).

Step 3: Apply the Formula

Substitute \( a = x^2 \) and \( b = 3y \) into the formula:\[ a - b = x^2 - 3y \]\[ a^2 + ab + b^2 = (x^2)^2 + x^2(3y) + (3y)^2 = x^4 + 3x^2y + 9y^2 \]

Step 4: Write the Final Factorization

Combine these results to factor the original polynomial:\[ x^6 - 27y^3 = (x^2 - 3y)(x^4 + 3x^2y + 9y^2) \].

Key Concepts

Difference of Cubes

The concept of the difference of cubes is a key technique in algebra used for factoring specific types of polynomials. It involves expressions that can be written in the form of \(a^3 - b^3\). These can be factored using a special formula. Recognizing the form \(a^3 - b^3\) allows us to use this formula to break down the expression into simpler parts. In our original exercise, the polynomial \(x^6 - 27y^3\) is indeed a difference of cubes. Here, \(a = x^2\) and \(b = 3y\). Understanding this pattern helps greatly in simplifying complex expressions.
Key aspects of the difference of cubes:

• The formula used is \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\).
• It requires identifying the perfect cube terms in the original expression.
• Once terms \(a\) and \(b\) are defined, the expression can be factored cleanly.

Practicing recognizing and applying the difference of cubes can significantly enhance your algebra problem-solving skills.

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into simpler components, or 'factors,' that can be multiplied together to give the original polynomial. Factorization simplifies the expression and is essential for solving polynomial equations. Especially, when the polynomial conforms to recognizable patterns like the difference of cubes or squares, factorization can simplify work immensely.
Steps in polynomial factorization:

• Identify patterns such as differences of squares or cubes that apply.
• Apply the correct factorization formula to break down the expression.
• Combine these factors, which often leads to easier solutions or simplifications of algebraic problems.

For the polynomial \(x^6 - 27y^3\), recognizing it as a difference of cubes allowed us to express it as \((x^2 - 3y)(x^4 + 3x^2y + 9y^2)\). Becoming proficient in such techniques is crucial for higher algebraic learning.

Algebraic Expressions

Algebraic expressions form the backbone of algebra. They involve variables, numbers, and arithmetic operations like addition, subtraction, multiplication, and division. Understanding algebraic expressions is critical as they allow mathematicians and students alike to express and solve real-world problems in a generalized form.
Core components of understanding and working with algebraic expressions:

• They can be simplified, expanded, and factored.
• The order of operations dictates how expressions are manipulated: parentheses first, then exponents, followed by multiplication and division, and finally addition and subtraction.
• Factoring, as demonstrated in our exercise, is a way to "open up" expressions to simplify solving or understanding them better.

By practicing the manipulation of algebraic expressions, students can advance their problem-solving skills and prepare for more complex algebraic topics.