Problem 78 - College Algebra

Question

In Exercises \(57-84\), factor completely, or state that the polynomial is prime. $$x^{2}-10 x+25-36 y^{2}$$

Step-by-Step Solution

Answer

The polynomial \(x^{2}-10 x+25-36 y^{2}\) completely factored is \((x-5-6y)(x-5+6y) \)

Step 1: Recognizing the perfect square trinomial

The first part of the polynomial \(x^{2}-10 x+25\) is a perfect square trinomial. This can be written as \((x-5)^{2}\). This is derived from the formula \(a^2 - 2ab + b^2 = (a - b)^2 \)

Step 2: Recognizing the difference of squares

The second part of the polynomial \(-36y^{2}\) along with the perfect square trinomial \(x^{2}-10 x+25\) which we have already written as \((x-5)^{2}\) form a difference of squares. The formula for difference of squares is \(a^2 - b^2 = (a - b)(a + b)\). Here, \(a = (x-5)\) and \(b = 6y\).

Step 3: Factoring the difference of squares

Applying the difference of squares formula, we have \((x-5)^2 - (6y)^2\) which gives us \((x-5-6y)(x-5+6y) \)

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special kind of polynomial that takes the form of \(a^2 - 2ab + b^2\). This expression simplifies to \((a - b)^2\). Rearranging and recognizing polynomials as perfect square trinomials can make factoring them incredibly simple.
The exercise has a trinomial: \(x^2 - 10x + 25\). Notice how it fits the form of a perfect square trinomial:

• Here, \(a = x\) and \(b = 5\).
• The expression can be rewritten as \((x - 5)^2\).

This identification helps simplify otherwise complex expressions, making them more manageable to work with, by transforming them into a perfect square.

Difference of Squares

The concept of a difference of squares involves expressions that follow the form \(a^2 - b^2\). These can be factored using the formula \((a - b)(a + b)\).
In our exercise, after recognizing the perfect square trinomial \((x - 5)^2\), we have:

• Expression: \((x - 5)^2 - 36y^2\)
• \(a = (x - 5)\)
• \(b = 6y\)

Thus, the expression can be seen as a difference of squares:- Substituting the values, we get \((x - 5 - 6y)(x - 5 + 6y)\).This powerful technique simplifies complex expressions into two binomial factors, making solving polynomial equations more straightforward.

Factoring Techniques

Factoring techniques are used to break down polynomials into a product of simpler expressions or factors, and they are essential for simplifying and solving polynomial equations.
Several important techniques include:

• Recognizing Special Polynomials: Identifying perfect square trinomials and differences of squares guiding towards easier factoring.
• Using Formulas: Such as \(a^2 - 2ab + b^2 = (a - b)^2\) for perfect square trinomials and \(a^2 - b^2 = (a - b)(a + b)\) for differences of squares.
• Simplifying Expressions: Turning complex expressions into more recognizable forms.

The exercise provides a clear demonstration of these techniques by first breaking down the perfect square trinomial and then applying the difference of squares formula, simplifying the expression into a product of factors.