Problem 86 - College Algebra

Question

Factor completely, or state that the polynomial is prime. $$ x^{2}-10 x+25-36 y^{2} $$

Step-by-Step Solution

Answer

The answer is \((x-5+6y)\) * \((x-5-6y)\)

Step 1: Recognize the Quadratic Part

First, you should recognize that the first three terms of the equation form a perfect square, in this case \( (x-5)^2\). So rewrite the expression as \((x-5)^2-36y^2\)

Step 2: Recognize the Difference of Squares

Second, the given polynomial can be seen as a difference of squares, a common form when factoring. The structure is \(A^2 - B^2\), which can be rewritten as \((A+B)\) * \((A-B)\). In this case, \(A=(x-5)\) and \(B=6y\).

Step 3: Apply Factoring Rule for Difference of Squares

Finally, apply the factoring rule for differences of squares. The factored form will be \((x-5+6y)\) * \((x-5-6y)\).

Key Concepts

Difference of Squares

In algebra, the difference of squares is a common method used for factoring. A polynomial is called a "difference of squares" when it can be expressed in the form \(A^2 - B^2\). This form can be factored into \((A + B)(A - B)\). It is a straightforward and very useful technique, especially because it can simplify expressions quickly.
To recognize a difference of squares:

• Look for two terms separated by a subtraction sign.
• Ensure both terms are perfect squares.

For instance, in the expression given \((x-5)^2 - 36y^2\):

• The term \((x-5)^2\) acts as \(A^2\).
• The term \(36y^2\) acts as \(B^2\), where \(B = 6y\).

Apply the formula to factor it as \((x-5 + 6y)(x-5 - 6y)\). This simple rewrite can make solving or simplifying an equation much easier!

Quadratic Expressions

Quadratic expressions are polynomials of the degree two, usually in the form of \(ax^2 + bx + c\). These expressions are fundamental in algebra and are often seen in various forms, including factored forms and vertex forms.
To factor a quadratic expression:

• Identify terms like \(x^2\), coefficient of \(x\) (linear term), and constant (c).
• Look for patterns or forms such as perfect square trinomials or differences of squares.

In the exercise, the first three terms \(x^2 - 10x + 25\) represent a quadratic expression. Recognizing these can help in identifying how to factor them further. Here, it was identified as a perfect square trinomial, \((x - 5)^2\), which is an important stepping stone in the factoring process.

Perfect Square Trinomials

A perfect square trinomial is a specific type of quadratic expression that can be expressed as the square of a binomial. It takes the form of \((a+b)^2\), which expands to \(a^2 + 2ab + b^2\). Recognizing such patterns can simplify many algebra problems.
Identifying a perfect square trinomial involves:

• Examining the structure: \(x^2 + 2xy + y^2\).
• Looking for a squared term, a twice-multiplied term, and another squared term.

In our exercise, \(x^2 - 10x + 25\) is rewritten as \((x-5)^2\). Observe how the middle term, \(-10x\), fits the pattern of \(-2\cdot x\cdot 5= -10x\). This characteristic allows us to simplify the process of factoring the larger expression efficiently.