Problem 93 - Algebra and Trigonometry with Analytic Geometry

Question

Factor the polynomial. $$ a^{6}-b^{6} $$

Step-by-Step Solution

Answer

The polynomial factors as \((a-b)(a^2+ab+b^2)(a+b)(a^2-ab+b^2)\).

Step 1: Recognize the Difference of Squares

The expression \( a^6 - b^6 \) is a difference of two squares, as it can be rewritten as \( (a^3)^2 - (b^3)^2 \). This is a specific form known as the difference of squares, which can be factored using the identity \( x^2 - y^2 = (x - y)(x + y) \).

Step 2: Factor Using the Difference of Squares

Apply the difference of squares formula: \((a^3)^2 - (b^3)^2 = (a^3 - b^3)(a^3 + b^3)\). This results in two binomial expressions.

Step 3: Factor the Difference of Cubes

Now, focus on \( a^3 - b^3 \), which is a difference of cubes. Use the difference of cubes formula: \( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \).

Step 4: Factor the Sum of Cubes

Similarly, look at \( a^3 + b^3 \), which is a sum of cubes. Use the sum of cubes formula: \( a^3 + b^3 = (a+b)(a^2 - ab + b^2) \).

Step 5: Write the Completely Factored Form

Combine everything. The completely factored form of \( a^6 - b^6 \) is:\((a-b)(a^2+ab+b^2)(a+b)(a^2-ab+b^2)\).

Key Concepts

Difference of Squares

The difference of squares is a specific algebraic identity represented by the formula \( x^2 - y^2 = (x - y)(x + y) \). It is applicable when you have two terms, each of which is a perfect square, separated by a subtraction sign. The original polynomial \( a^6 - b^6 \) can be rewritten to fit this format: \( (a^3)^2 - (b^3)^2 \). In this setup, you see each component squared, making it easier to apply the identity.

• Step 1: Identify perfect squares in the problem.
• Step 2: Recognize the subtraction, signaling a difference.
• Step 3: Apply the difference of squares identity \((x-y)(x+y)\).

Breaking the polynomial into \((a^3 - b^3)(a^3 + b^3)\) is an effective application of the difference of squares formula.

Difference of Cubes

The difference of cubes involves breaking down expressions like \( a^3 - b^3 \) using the specific formula: \( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \). This formula is essential when a polynomial contains cubes of variables. In the expanded factorization sequence from \( a^6 - b^6 \), focus shifts to the expression \( a^3 - b^3 \).

• Step 1: Confirm both terms are cubes.
• Step 2: Use the difference of cubes formula.

By implementing the difference of cubes identity, \( a^3 - b^3 \) transforms into \( (a-b)(a^2 + ab + b^2) \), which further simplifies the polynomial.

Sum of Cubes

Addressing sums of cubes follows a similar structure to differences of cubes, but it applies to formats like \( a^3 + b^3 \). Their factorization uses a distinct formula: \( a^3 + b^3 = (a+b)(a^2 - ab + b^2) \). Spotting this form in polynomial expressions makes it easier to manage complex algebraic expressions.

• Step 1: Confirm presence of cubes with a sum.
• Step 2: Apply the sum of cubes formula directly.

In \( a^6 - b^6 \), after recognizing the sum part \( a^3 + b^3 \), using this identity factors the expression to \( (a+b)(a^2 - ab + b^2) \), completing one half of the factorization journey.

Binomial Expressions

Binomial expressions are algebraic expressions containing exactly two terms. They often appear in polynomial factorization processes. Using the combination of identities like difference or sum of squares/cubes results in the creation or simplification of binomials.

• Examples include expressions like \(a+b\) or \(a-b\).
• They can often be multiplied to further factor polynomial expressions.

In the completely factored form of \( a^6 - b^6 \), you see multiple binomials appearing as a result of applying different factorization formulas, such as \((a-b), (a+b), (a^2+ab+b^2), (a^2-ab+b^2)\). Watching how these binomial expressions emerge through step-by-step factorization helps understand polynomial simplification.