Problem 144 - College Algebra with Modeling and Visualization

Question

Factor the expression completely. \(9 y^{2}-16 z^{2}\)

Step-by-Step Solution

Answer

The expression factors to \((3y - 4z)(3y + 4z)\).

Step 1: Identify the Form

Recognize that the expression \(9y^2 - 16z^2\) is a difference of squares. The difference of squares formula is \(a^2 - b^2 = (a - b)(a + b)\).

Step 2: Find Square Roots

Determine the square roots of each term in the expression. \(9y^2\) can be written as \((3y)^2\), and \(16z^2\) can be written as \((4z)^2\).

Step 3: Apply the Difference of Squares Formula

Substitute the square roots into the difference of squares formula: \((3y)^2 - (4z)^2 = (3y - 4z)(3y + 4z)\).

Step 4: Verify the Factorization

Multiply the factors \((3y - 4z)\) and \((3y + 4z)\) to check that they reproduce the original expression: \((3y - 4z)(3y + 4z) = 9y^2 - 16z^2\). This confirms the factorization is correct.

Key Concepts

Difference of Squares

The difference of squares is a fascinating concept in algebra that simplifies factorization. This technique is used to identify and factor certain types of expressions. The core formula for the difference of squares is:

• \(a^2 - b^2 = (a - b)(a + b)\)

This formula reveals that the expression is composed of two squares — that means each term must be a perfect square. The whole expression equals the product of two binomials.
To apply this concept appropriately, you have to recognize both terms as squares. For example, in the expression \(9y^2 - 16z^2\), the terms \(9y^2\) and \(16z^2\) are squares of \(3y\) and \(4z\) respectively.
Identifying these square terms helps in rewriting the difference of squares into its factored form, where it becomes \((3y - 4z)(3y + 4z)\). This reveals the simplicity of using this pattern.
Understanding and remembering the difference of squares can greatly simplify the task of solving or factoring polynomial expressions.

Factoring

Factoring is a crucial skill in algebra, as it transforms expressions into multiplied elements, making them easier to solve or manipulate. In general, factoring involves breaking down a complex expression into simpler terms or the product of simpler polynomials.
In the context of the difference of squares, factoring is straightforward once you recognize the expression's structure. Using the example expression \(9y^2 - 16z^2\), once you identify that it fits the \(a^2 - b^2\) format, the factoring process becomes:

• Identify each part as a square: \(9y^2 = (3y)^2\) and \(16z^2 = (4z)^2\).
• Use the difference of squares formula to factor: substitute these terms to form the factors \((3y - 4z)\) and \((3y + 4z)\).
• Verify by expanding to ensure the factored expression equals the original expression.

Factoring is not only about finding factors; it’s also about verifying correctness, as demonstrated above. This verification involves multiplying the factors back to check if they create the original expression, affirming the factorization's correctness.

Polynomials

Polynomials are algebraic expressions composed of variables, coefficients, and the operation of addition, subtraction, and multiplication. Understanding polynomials is fundamental to algebra as they appear frequently in various forms and complexities.
A polynomial like \(9y^2 - 16z^2\) involves recognizing patterns that can simplify operations. Here, distinction between monomials (single terms like \(9y^2\)) and binomials (expressions like \(9y^2 - 16z^2\)) is vital.
In

• Polynomials usually are depicted in a standard form, such as \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\), where powers are in descending order.
• Each component in a polynomial takes part in processes like addition, subtraction, multiplication, and factorization.

Understanding polynomials enables you to manipulate expressions and solve equations more effectively. Recognizing patterns, like the difference of squares within polynomials, shortens complex operations and leads to accurate, efficient solutions.