Find the GCF of the terms of each polynomial. 14 z^4-42 z^3+21 z^2

Name: Problem 20, Chapter 8: Algebra 1 Common Core
Uploaded: 2020-03-29T04:35:19-08:00
Duration: 4 min 27 s
Channel: Jason Taylor-Pestell
Description: Find the GCF of the terms of each polynomial. 14 z^4-42 z^3+21 z^2

Find the GCF of the terms of each polynomial. 14 z^4-42...

Key Concepts

Common Factors in Coefficients

When finding the GCF, it is essential to identify the highest number that divides each numerical coefficient evenly. This step reduces the expression before considering any shared factors in the variable parts.

Common Factors in Variables and Exponents

In terms with variables, the common factor is determined by selecting the variable raised to the lowest exponent found among all terms. This ensures that the extracted factor divides evenly into each term's variable component.

Polynomial Structure

A polynomial is an expression composed of sums or differences of terms, where each term is a product of a numerical coefficient and a variable raised to a nonnegative integer power. Understanding the structure of polynomials is crucial for applying various techniques such as factoring.

Greatest Common Factor (GCF)

The GCF is the largest expression that divides each term of an algebraic expression without leaving a remainder. It involves identifying both the numerical factors and the common variable components with their smallest exponents across all terms.

Factoring Techniques

Factoring involves rewriting a polynomial as a product of simpler expressions. One of the first steps in factoring is extracting the GCF from all terms, which simplifies the expression and can make further factoring steps either obvious or simpler.

Transcript

00:01 All right.

00:01 So in this problem, we are looking for the greatest common factor of the three terms that we see in this polynomial.

00:06 I want to begin by just making a note that when i set up this problem, i originally had 4 z squared, or sorry, z to the 4th.

00:14 And it's really easy to, when you are starting to solve a problem, to miss one number, one digit, one exponent.

00:22 And so i just want to encourage you to always double check to make sure that the problem that you have in front of you is the problem that you're supposed to.

00:30 To be solving, that you didn't copy anything incorrectly like i did originally and then noticed it right before starting this video.

00:38 So the strategy that i'm going to be using is called double division, and double division looks for the common factors in two different terms.

00:48 In this case, we're extending it to triple division, meaning we're looking for the common terms, or sorry, the common factors within these three different terms.

00:58 So to begin, i'm looking at the coefficients 14, 42, and 21.

01:05 And i notice that 14 is the same thing as 2 times 7, 42 is the same thing as 6 times 7, and 21 is 3 times 7.

01:15 And so 7 is a common number that i notice in all three of them.

01:18 So i'm going to put that common factor of 7 outside.

01:22 That means that i'm factoring a 7 out from each of the terms, and the result will be put down below.

01:29 So if i just focus on the numbers right now, factoring a 7 out from 14 means i'm left with 2.

01:36 Factoring a 7 out from negative 42 means i'm left with negative 6.

01:43 And factoring a 7 out from positive 21 means i'm left with positive 3.

01:50 Next, i'm going to focus on the variables.

01:53 So i notice that all three of them have a z.

01:56 And they actually have multiple factors of z.

01:58 This one has four factors.

02:01 The middle term has three factors, and the last term has two factors of z.

02:06 Common factors can only be the ones that they share, that all of them have.

02:11 And so it's the minimum number that you see, which in this case is the last one's z squared.

02:16 So i'm going to pull out a z squared or two factors of z...

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