Problem 153 Factor each polynomial completel... [FREE SOLUTION]

91影视

Chapter 5: Problem 153

Factor each polynomial completely. $$9 a^{2}-z^{2}$$

Short Answer

Expert verified

(3a + z)(3a - z)

Step by step solution

Identify the Polynomial Structure

The polynomial given is in the form of a difference of squares: \[9a^2 - z^2\]

Rewrite as Squares

Rewrite each term in the polynomial as squares: \[9a^2 = (3a)^2\] and \[z^2 = z^2\]. Therefore, the polynomial can be rewritten as: \[(3a)^2 - z^2\]

Apply the Difference of Squares Formula

Use the difference of squares formula, which states \[a^2 - b^2 = (a + b)(a - b)\]. Here, let \[a = 3a\] and \[b = z\]. Applying the formula, we get: \[(3a + z)(3a - z)\]

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Difference of Squares

The difference of squares is a specific type of polynomial and is one of the key concepts in factoring. This type of polynomial occurs when we have two square terms separated by a minus sign, like $a^2 - b^2$. In the equation $9a^2 - z^2$, it's clear that both $9a^2$ and $z^2$ are square terms.

To identify and factor a difference of squares, remember these steps:

Check if both terms are perfect squares.
Ensure there is a subtraction (the minus sign) between them.
Apply the difference of squares formula: $a^2 - b^2 = (a + b)(a - b)$.

By understanding and recognizing this pattern, you can quickly factor many polynomials of this form.

Polynomial Structure

Understanding the structure of polynomials is crucial for successful factoring. Polynomials can come in various forms, and knowing the type can simplify the factoring process.

For the polynomial $9a^2 - z^2$, we need to look at each term:

The term $9a^2$, which is a perfect square because $9a^2 = (3a)^2$.
The term $z^2$, which is also a perfect square because $z^2 = z^2$.

Once we recognize these terms as squares, we can rewrite the polynomial as $(3a)^2 - z^2$.

Recognizing the polynomial structure helps in determining the right method to factor it. In this case, it led us to identify it as a difference of squares.

Factoring Formulas

Factoring formulas are essential tools in algebra. They provide shortcuts to rewrite expressions in a simpler form. One of the most important formulas is the difference of squares formula: $a^2 - b^2 = (a + b)(a - b)$.
This formula tells us that any expression that is the difference of two squares can be factored into the product of two binomials.

To apply this formula to our example:
For $9a^2 - z^2$:

Identify $a = 3a$ and $b = z$.
Apply the formula: $ (3a)^2 - z^2 = (3a + z)(3a - z)$.

This step-by-step approach simplifies the polynomial, making it easier to handle in calculations or further manipulations. Factoring using formulas is a powerful technique, especially for complex algebraic expressions.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Factor each polynomial completely. $$9 a^{2}-z^{2}$$

Short Answer

Step by step solution

Identify the Polynomial Structure

Rewrite as Squares

Apply the Difference of Squares Formula

Key Concepts

Difference of Squares

Polynomial Structure

Factoring Formulas

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Applied Mathematics

Geometry

Theoretical and Mathematical Physics

Mechanics Maths

Pure Maths

Study anywhere. Anytime. Across all devices.