Problem 118 - Elementary Algebra

Question

Factor each expression. \(x^{2}-16 x y+64 y^{2}\)

Step-by-Step Solution

Answer

The factored expression is \( ( x - 8 y )^2 \).

Step 1: Identify the structure of the quadratic expression

Observe that the given expression is a quadratic expression of the form \( ax^2+bx+c \). Here, the expression is \( x^2-16xy+64y^2 \).

Step 2: Check if it is a perfect square trinomial

A perfect square trinomial has the form \( (a x + b y)^2 \). Express the given quadratic as a trinomial square \( ( x - 8y )^2 \) if possible.

Step 3: Factor the expression

Simplify \( ( x - 8y )^2 \) to obtain the factored form of the original expression.

Key Concepts

Perfect Square Trinomials

A perfect square trinomial is a type of quadratic expression. It can always be written as the square of a binomial. In general, a perfect square trinomial looks like \((a x + b y)^2\). This form expands to \((a x)^2 + 2 a b x y + (b y)^2\).

If you compare this with the quadratic expressions you encounter, you will see the pattern.\

Quadratic Equations

Quadratic equations are polynomials of degree 2. They have the general form \( a x^2 + b x + c = 0 \).

When solving quadratic equations, factoring them is one useful method. Factoring changes the quadratic into a product of two linear expressions. This can make it easier to solve for the variable involved.

Quadratics can also appear with more than one variable, like in the original exercise \(x^{2}-16 x y+64 y^{2} \).

Algebraic Factoring

Algebraic factoring is breaking down a complex expression into simpler expressions that when multiplied give the original one.

It is particularly useful for solving quadratic equations and simplifying expressions. There are several types of factoring methods, including:

• Factoring out the greatest common factor (GCF).
• Factoring by grouping.
• Factoring perfect square trinomials.
• Factoring the difference of squares.

With consistent practice, recognizing patterns and applying the correct method becomes easier.

In the exercise, we used the method of factoring perfect square trinomials to simplify the given quadratic expression \( x^{2}-16 x y+64 y^{2} \) into \( (x-8y)^{2} \).