Problem 41 - Algebra for College Students

Question

Factor each of the following polynomials completely. Indicate any that are not factorable using integers. Don't forget to look first for a common monomial factor. $$4 x^{2}-64 y^{2}$$

Step-by-Step Solution

Answer

The complete factorization is \(4(x + 4y)(x - 4y)\).

Step 1: Look for a Common Monomial Factor

Examine the polynomial \(4x^2 - 64y^2\) for any common monomial factors in each term. In this case, both terms have a common factor of 4. Factor out the 4: \[4(x^2 - 16y^2)\].

Step 2: Recognize the Difference of Squares

Identify that \(x^2 - 16y^2\) is a difference of squares. A difference of squares follows the formula \(a^2 - b^2 = (a+b)(a-b)\). Here, \(a = x\) and \(b = 4y\), since \((4y)^2 = 16y^2\).

Step 3: Apply the Difference of Squares Formula

Using the formula from Step 2, factor \(x^2 - 16y^2\) as \((x + 4y)(x - 4y)\).

Step 4: Combine the Factors

Combine the factors found in the previous steps. The complete factorization of the polynomial \(4x^2 - 64y^2\) is \[4(x + 4y)(x - 4y)\].

Key Concepts

Common Monomial Factor

In the world of algebra, simplifying polynomials often involves looking for common elements. A **common monomial factor** refers to a term that can be factored out from each part of a polynomial. This is a key step because it simplifies the polynomial and prepares it for further factoring.
Let's take an example: if we have a polynomial like \(4x^2 - 64y^2\), both terms in this expression have a number that divides evenly into them, which is 4. Therefore, 4 can be considered a common monomial factor.
To factor it out, you divide each term by this common factor and place it outside a set of parentheses. So, \(4x^2 - 64y^2\) becomes \(4(x^2 - 16y^2)\). This simplification reveals the structure of the polynomial, making it easier to factor further.
Here are some quick tips to spot and factor out a common monomial factor:

• Look for the greatest common factor (GCF) shared by all terms.
• Ensure that the GCF divides each term completely.
• Remember that factoring does not change the value of the polynomial, it just rewrites it in a simpler form.

Recognizing and extracting the common monomial factor streamlines the factoring process and sets a strong foundation for subsequent steps.

Difference of Squares

When working with polynomials, recognizing patterns is vital. A **difference of squares** is a specific type of pattern where a polynomial is expressed as the difference between two perfect squares. It follows the form \(a^2 - b^2\). This is important because it can be factored into two binomials, \((a+b)(a-b)\).
Consider our simplified polynomial from before: \(x^2 - 16y^2\).
This is a classic example of a difference of squares:

• \(x^2\) is the square of \(x\)
• \(16y^2\) is actually \((4y)^2\)

Hence, we can rewrite \(x^2 - 16y^2\) as \( (x + 4y)(x - 4y) \).
Here’s why recognizing the difference of squares is so valuable:

• It transforms the polynomial into a simpler, factorable form.
• The pattern \(a^2 - b^2 = (a + b)(a - b)\) is universal and can be applied to any difference of squares.

By identifying this structure, you simplify further factoring processes and solve algebraic expressions efficiently.

Polynomial Simplification

Simplifying polynomials is about transforming a polynomial into its most manageable form. This often means breaking it down into components that are easier to work with, like factors.
After factoring out a common monomial and recognizing special patterns such as the difference of squares, you achieve a simplified expression that maintains the same value. For our polynomial \(4x^2 - 64y^2\), after applying these techniques, it ultimately simplifies to \[4(x + 4y)(x - 4y)\].
Simplification makes verifying mathematical truths and solving equations much easier because:

• It reduces the complexity of algebraic expressions.
• It allows for straightforward calculation of polynomial values for given variables.
• It helps in identifying properties or characteristics of the polynomial more easily.

Next time you face a complex polynomial, remember these concepts to simplify your work and improve your mathematical efficiency!