Problem 64 Factor completely. Identify any ... [FREE SOLUTION]

91影视

Elementary Algebra

Laura Bracken, Ed Miller

$Math Studyset 91影视 Explanations$ Math

1 Edition

Chapter 6: Problem 64

Factor completely. Identify any prime polynomials. $$ 5 k y^{2}-10 y^{2} $$

Short Answer

Expert verified

The factored form is $ 5y^2(k - 2) $. $ k - 2 $ is a prime polynomial.

Step by step solution

Identify the GCD

Determine the greatest common divisor (GCD) of the coefficients and the variables in the terms. In this case, both terms have a common factor of 5 and a common variable factor of $ y^2 $.

Factor out the GCD

Factor out the GCD from the polynomial. The GCD is $ 5y^2 $. When you factor it out, the expression becomes: \[ 5y^2(k - 2) \]

Check for prime polynomial

Check if the remaining factor $ k - 2 $ can be factored further. Since $ k - 2 $ is a linear polynomial without further factorization, it is a prime polynomial.

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.

Greatest Common Divisor (GCD)

When factoring polynomials, finding the Greatest Common Divisor (GCD) is a crucial first step. The GCD is the highest number that divides all terms of a polynomial exactly. In the polynomial given, \[ 5ky^2 - 10y^2 \], we identify the GCD by examining both the coefficients and the variable parts. For the coefficients, the numbers 5 and 10 share a GCD of 5. For the variables, the common factor is $ y^2 $. Hence, the GCD of the entire polynomial is $ 5y^2 $. This means that both terms can be divided by $ 5y^2 $.

Prime Polynomials

Once you factor out the GCD, you need to check if the resulting polynomial can be factored further. Let's look at the expression after factoring out the GCD:\[ 5y^2(k - 2) \]. Here, the factor inside the parentheses is $ k - 2 $. A polynomial is considered 'prime' when it cannot be factored further over the set of integers. In our case, $ k - 2 $ is a linear polynomial and cannot be factored any further. Thus, $ k - 2 $ is a prime polynomial. Identifying prime polynomials is important because it tells us that we have factored the polynomial completely.

Factoring Techniques

Factoring is a technique used to simplify polynomials by breaking them down into simpler 'factors' that, when multiplied together, give the original polynomial. Here are the steps to follow when factoring a polynomial, illustrated with our example:

Identify the GCD: As previously discussed, find the greatest common factor of all terms. In our case, it was $ 5y^2 $.
Factor out the GCD: Divide each term of the polynomial by the GCD and extract it. The polynomial becomes $ 5y^2(k - 2) $.
Check for further factorization: Examine each factor to see if they can be factored more. For our example, $ k - 2 $ is a linear factor and prime.

These steps simplify the polynomial and make it easier to work with. Consistent practice with these techniques helps in identifying factorable and prime polynomials efficiently.

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 completely. Identify any prime polynomials. $$ 5 k y^{2}-10 y^{2} $$

Short Answer

Step by step solution

Identify the GCD

Factor out the GCD

Check for prime polynomial

Key Concepts

Greatest Common Divisor (GCD)

Prime Polynomials

Factoring Techniques

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Theoretical and Mathematical Physics

Decision Maths

Geometry

Pure Maths

Statistics

Study anywhere. Anytime. Across all devices.