Problem 75 - Elementary Algebra

Question

For the following problems, factor the polynomials, if possible. $$ 81 a^{8} b^{12} c^{10}-25 x^{20} y^{18} $$

Step-by-Step Solution

Answer

Answer: The factored form of the given polynomial is $$(9 a^{4} b^{6} c^{5} + 5 x^{10} y^9)(9 a^{4} b^{6} c^{5} - 5 x^{10} y^9)$$

Step 1: Identify the common factors

There are no common factors between the two terms in the polynomial since they have different variables. As a result, we cannot find a common factor to work with.

Step 2: Check for difference of square

Let's check if the given polynomial has a difference of squares form, which allows us to use the formula: $$ a^2 - b^2 = (a + b)(a - b) $$ The given polynomial expression is: $$ 81 a^{8} b^{12} c^{10} - 25 x^{20} y^{18} $$ We notice that both terms are perfect squares: $$ (9 a^{4} b^{6} c^{5})^2 - (5 x^{10} y^9)^2 $$

Step 3: Apply the difference of squares formula

We can rewrite the given polynomial as the difference of squares, applying the formula from step 2: $$ (9 a^{4} b^{6} c^{5} + 5 x^{10} y^9)(9 a^{4} b^{6} c^{5} - 5 x^{10} y^9) $$ The factored form of the given polynomial is: $$ (9 a^{4} b^{6} c^{5} + 5 x^{10} y^9)(9 a^{4} b^{6} c^{5} - 5 x^{10} y^9) $$

Key Concepts

Difference of Squares

The difference of squares is a very useful concept in algebra, particularly when factoring polynomials. It refers to a specific pattern where a polynomial can be expressed as one square minus another square. In mathematical terms, this can be written as \(a^2 - b^2\). This particular structure allows us to factor the polynomial using a straightforward formula:

• \( a^2 - b^2 = (a + b)(a - b) \)

This formula states that the difference of two squares can be broken down into the product of two binomials, where one is the sum of the square roots, and the other is the difference of the square roots.
In our example, "81\(a^{8}b^{12}c^{10}\) - 25\(x^{20}y^{18}\)", each term is a perfect square because \((9a^4b^6c^5)^2\) is \(81a^{8}b^{12}c^{10}\) and \((5x^{10}y^9)^2\) is \(25x^{20}y^{18}\). This identifies the expression as a difference of squares, and we can apply our formula directly to get a factored form consisting of two binomials.

Perfect Squares

Perfect squares are terms in polynomials that can be expressed as the square of some other expression. When you encounter a situation where terms look like perfect squares within a polynomial, you can often simplify and factor the expression more easily.

• The expression \(a^2 \) is a perfect square because it is the square of \(a\).
• Similarly, \(b^2\) is a perfect square as it equals \(b\) times \(b\).

In our polynomial example, "81\(a^{8}b^{12}c^{10}\) - 25\(x^{20}y^{18}\)", we see that each term is a perfect square:

• \(81a^{8}b^{12}c^{10}\) is the square of \(9a^4b^6c^5\).
• \(25x^{20}y^{18}\) is the square of \(5x^{10}y^9\).

Recognizing these patterns is crucial because if each term in the polynomial is a perfect square, the polynomial can potentially be factored as a difference of squares or some other form. This simplification makes it much easier to solve or further manipulate the expression.

Polynomial Expressions

Polynomial expressions consist of variables and coefficients structured with operations like addition, subtraction, and multiplication but not division by a variable. A polynomial is generally expressed in the form \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where \(a_n\), \(a_{n-1}\), ..., \(a_0\) are constants and \(x\) is a variable. Each complete form, or term, of the expression has a coefficient and can be raised to a power.
Factoring polynomials involves rewriting the expression as a product of its factors, which are simpler polynomial expressions or even constants. The goal of factoring is often to simplify the polynomial into a set of problems or solutions that are easier to handle.
In this particular problem, we are working with the polynomial "81\(a^{8}b^{12}c^{10}\) - 25\(x^{20}y^{18}\)", which is structured as a difference of squares.

• The two main types of polynomials involved are the perfect squares that when combined in specific ways (additive and subtractive), allow us to factor using the difference of squares theorem.

Understanding the structure of polynomial expressions is fundamental to mastering algebraic manipulation, such as factoring by recognizing patterns like perfect squares and the difference of squares.