Problem 59 - Elementary Algebra

Question

For the following problems, factor the binomials. $$ 49 x^{2} y^{4} z^{6}-64 a^{4} b^{2} c^{8} d^{10} $$

Step-by-Step Solution

Answer

Question: Factor the given binomial: $$ (49x^4y^4z^6 - 64a^4b^2c^8d^{10}) $$ Answer: The factored form of the given binomial is: $$ (7x^2y^2z^3 + 8a^2b\sqrt{c^8}d^5)(7x^2y^2z^3 - 8a^2b\sqrt{c^8}d^5) $$

Step 1: Identify the common factors

In this case, there are no common factors that can be extracted from the coefficients and variables of the given binomial.

Step 2: Rewrite the binomial as a difference of squares

Express the given binomial as \(a^2 - b^2\), where \(a = 7x^2y^2z^3\) and \(b = 8a^2b\sqrt{c^8}d^5\). So the binomial can be written as: $$ (7x^2y^2z^3)^2 - (8a^2bcdot\sqrt{c^8}d^5)^2 $$

Step 3: Apply the difference of squares formula

Using the difference of squares formula \(a^2 - b^2 = (a + b)(a - b)\), apply it to the binomial: $$ (7x^2y^2z^3 + 8a^2b\sqrt{c^8}d^5)(7x^2y^2z^3 - 8a^2b\sqrt{c^8}d^5) $$

Step 4: Simplify the factored result

Since there are no further common factors among the terms in the parentheses, the simplified factored result is: $$ (7x^2y^2z^3 + 8a^2b\sqrt{c^8}d^5)(7x^2y^2z^3 - 8a^2b\sqrt{c^8}d^5) $$

Key Concepts

Difference of Squares

The difference of squares is a powerful tool in algebra for factoring specific types of binomials. It applies when you have a binomial of the form \(a^2 - b^2\), which can be factored as \((a + b)(a - b)\). This formula works because multiplying \((a + b)\) by \((a - b)\) results in \(a^2 - b^2\).

In the exercise given, the expression \(49 x^{2} y^{4} z^{6}-64 a^{4} b^{2} c^{8} d^{10}\) can be rewritten to fit the difference of squares model:

• Identify \(a = 7x^2y^2z^3\), which squares to give \(49x^4y^4z^6\).
• Set \(b = 8a^2b\sqrt{c^8}d^5\), which squares to yield \(64a^4b^2c^8d^{10}\).

This allows us to express the binomial in terms of \(a^2 - b^2\).

Employing the difference of squares formula simplifies factoring these types of binomials without having to guess or check. It’s essential to recognize the structure of \(a^2 - b^2\) in algebraic expressions to apply this technique effectively.

Polynomial Factoring

Polynomial factoring involves breaking down a polynomial into simpler products that are multiplied together to give the original expression. This process is crucial for simplifying polynomials and solving polynomial equations.

The key steps in factoring include:

• Identifying a binomial or polynomial pattern that allows us to apply specific formulas, such as the difference of squares.
• Rewriting terms to reveal hidden patterns that match these formulas.
• Using methods like the greatest common factor (GCF) first, if applicable.
• Applying special formulas for binomials, such as \((a^2 - b^2)\), to achieve a factored result.

In the current problem, polynomial factoring is accomplished by using the difference of squares technique. Through recognition and application of the correct formula, we efficiently simplify the expression into a product of two binomials.

Algebraic Expressions

Working with algebraic expressions is a fundamental aspect of algebra. These expressions contain numbers, variables, and operations. In this context, understanding algebraic manipulation is key to solving and simplifying expressions or equations.

Key elements when dealing with algebraic expressions:

• Recognizing the types of polynomials, such as monomials, binomials, and trinomials.
• Understanding how to manipulate these expressions using basic operations: addition, subtraction, multiplication, and division.
• Applying formulas such as the difference of squares or perfect square trinomials to simplify expressions.
• Recognizing like terms to combine them effectively.

In the provided exercise, comprehending the algebraic expression \(49 x^{2} y^{4} z^{6}-64 a^{4} b^{2} c^{8} d^{10}\) meant identifying a structure that fits the difference of squares formula. This understanding allowed us to efficiently factor the expression, demonstrating the power of algebraic manipulation.