Factor Polynomials with GCMF Video

Summary

This video script offers a comprehensive guide to factoring polynomials by identifying the greatest common monomial factor (GCMF). It explains the process of finding the GCMF by examining both numerical coefficients and variable exponents, using examples like 6x + 4x^2 and 9x^3y^7 - 18xy^2 - 36y. The script demonstrates how to simplify polynomials by dividing each term by the GCMF and combining the results, resulting in factored forms that are easier to understand and solve.

Takeaways

• 🔍 The target of the script is to teach how to factor polynomials with the greatest common monomial factor (GCM).
• 📚 The GCM is defined as the highest possible value of a number or a combination of numbers and variables that is common in each term of a polynomial.
• 📝 The process of factoring polynomials involves three main steps: identifying the GCM, dividing each term by the GCM, and combining the results.
• 📉 For the polynomial 6x + 4x^2, the GCM of the numerical coefficients (6 and 4) is 2, and the GCM of the variables (x and x^2) is x to the least exponent, which is x.
• 📈 After finding the GCM, the script demonstrates dividing each term of the polynomial by the GCM, resulting in 3 and 2x, respectively.
• 📚 The script then combines the results to show the factored form of the polynomial as 2x(3 + 2x).
• 🔢 In the second example, the polynomial 9x^3y^7 - 18xy^2 - 36y, the GCM of the numerical parts (9, -18, -36) is 9, and the GCM of the variables is y squared.
• 📉 The script illustrates dividing each term of the polynomial by the GCM, 9y^2, to simplify it to x^3y^8 - 2x - 45.
• 📝 The final factored form of the second polynomial is 9y^2(x^3y^5 - 2x - 45).
• 👉 The script emphasizes the importance of finding the least exponent for the variable part of the GCM when variables have different powers.
• 📌 The examples provided in the script serve to illustrate the step-by-step process of factoring polynomials using the GCM method.

Q & A

What is the greatest common monomial factor (GCMF)?

The GCMF is the highest possible value of a number or a combination of numbers and variables which is common in each term of a polynomial.

What are the steps involved in factoring polynomials using the greatest common factor?

The steps are: 1) Find the greatest common monomial factor, 2) Divide each term in the polynomial by the GCMF, and 3) Combine the answers from steps one and two.

How do you determine the greatest common factor (GCF) of numerical coefficients in a polynomial?

You determine the GCF of the numerical coefficients by finding the largest number that divides evenly into all the coefficients of the terms in the polynomial.

What is the GCF of the variables in the polynomial 6x + 4x^2?

The GCF of the variables in 6x + 4x^2 is x, since x is the common variable with the least exponent.

What is the greatest common monomial factor for the polynomial 6x + 4x^2?

The GCMF for the polynomial 6x + 4x^2 is 2x, which is the product of the GCF of the numerical coefficients (2) and the GCF of the variables (x).

How do you simplify the polynomial 6x + 4x^2 after finding the GCMF?

After finding the GCMF of 2x, you divide each term by 2x, resulting in 3 and 2x, respectively, and then write the simplified polynomial as 2x(3 + 2x).

What is the GCF of the numerical parts of the polynomial 9x^3y^7 - 18xy^2 - 36y?

The GCF of the numerical parts (9, -18, -36) is 9.

How do you determine the GCF of the variables in the polynomial 9x^3y^7 - 18xy^2 - 36y?

The GCF of the variables is determined by the least exponent of the common variable present in all terms, which in this case is y squared (y^2).

What is the greatest common monomial factor for the polynomial 9x^3y^7 - 18xy^2 - 36y?

The GCMF for the polynomial 9x^3y^7 - 18xy^2 - 36y is 9y^2, considering both the GCF of the numerical parts and the GCF of the variables.

How do you simplify the polynomial 9x^3y^7 - 18xy^2 - 36y after finding the GCMF?

After finding the GCMF of 9y^2, you divide each term by 9y^2, resulting in x^3y^5, -2x, and -4, and then write the simplified polynomial as 9y^2(x^3y^5 - 2x - 4).

Why is it important to factor polynomials?

Factoring polynomials is important as it simplifies the expression, making it easier to solve equations, understand the roots, and perform various algebraic manipulations.

Outlines

00:00

📚 Factoring Polynomials with Greatest Common Monomial Factor

This paragraph introduces the concept of factoring polynomials by finding the greatest common monomial factor (GCMF). It explains that the GCMF is the highest common numerical factor and variable with the least exponent present in each term of the polynomial. The process involves three steps: identifying the GCMF by finding the greatest common factor (GCF) of the numerical coefficients and the least exponent of the common variable, dividing each term by the GCMF, and then combining the results to express the polynomial as a product of simpler terms. An example is given where the polynomial 6x + 4x^2 is factored by determining the GCMF as 2x, dividing each term by this factor, and combining them to get the final factored form of 3 + 2x.

05:01

🔍 Advanced Factoring with Greatest Common Monomial Factor

The second paragraph delves into a more complex example of factoring, focusing on a polynomial with both numerical and variable components. It outlines the process of determining the GCMF by identifying the GCF of the numerical coefficients and the least exponent of the common variable across all terms. The example given is the polynomial 9x^3y^7 - 18xy^2 - 36y, where the GCF of the numerical coefficients is 9, and the least exponent of the common variable y is squared. The GCMF is thus 9y^2. The polynomial is then divided by this GCMF, resulting in the factored form of x^3y^8 - 2xy - 45. This demonstrates the application of the GCMF in simplifying and factoring more complex polynomial expressions.

Keywords

💡Factor Polynomials

Factoring polynomials is the process of breaking down a polynomial into a product of simpler polynomials or factors. In the video, this concept is central to understanding the steps involved in simplifying expressions. For example, the polynomial '6x + 4x^2' is factored by identifying and dividing out the greatest common monomial factor, which in this case is '2x'.

💡Greatest Common Monomial Factor (GCMF)

The GCMF is the highest common factor of both the numerical coefficients and the variables in a polynomial, taking into account the lowest power of the variables. In the script, the GCMF is determined as '2x' for the polynomial '6x + 4x^2', which is then used to simplify the expression.

💡Greatest Common Factor (GCF)

The GCF refers to the largest number that divides two or more integers without leaving a remainder. In the context of the video, the GCF is used to find the common numerical coefficients in a polynomial, such as finding the GCF of 6 and 4, which is 2.

💡Monomial

A monomial is a single term algebraic expression, consisting of a product of numbers and variables raised to nonnegative integer powers. In the video, the concept of monomials is important for identifying the GCMF, as it involves looking at the individual terms of the polynomial.

💡Variable

A variable is a symbol, often a letter, that represents an unknown quantity in an algebraic expression. In the script, variables such as 'x' and 'y' are used in polynomials, and their exponents are considered when determining the GCMF.

💡Least Exponent

The least exponent refers to the smallest power to which a variable is raised in a polynomial. In the video, when determining the GCMF, the least exponent of the common variable is chosen to ensure the factor is as simple as possible, such as 'x' in '6x + 4x^2'.

💡Numerical Coefficient

Numerical coefficients are the numerical parts of a term in a polynomial that multiply the variables. In the video, finding the GCF of the numerical coefficients, like 6 and 4, is the first step in determining the GCMF.

💡Divide Each Term

This step in the factoring process involves dividing each term of the polynomial by the GCMF to simplify the expression. In the script, after finding '2x' as the GCMF, each term of '6x + 4x^2' is divided by '2x' to get the simplified factors.

💡Simplify

Simplifying a polynomial means reducing it to its simplest form by factoring out common terms. In the video, the process of simplifying is demonstrated by dividing the polynomial by its GCMF, resulting in a more manageable expression.

💡Polynomial

A polynomial is an algebraic expression consisting of a sum of terms, each term being a product of a numerical coefficient and a non-negative integer power of a variable. The video script discusses factoring polynomials, which is a method to break them down into simpler components.

💡Steps in Factoring

The steps in factoring are a systematic approach to simplifying polynomials. The video outlines these steps, starting with finding the GCMF, dividing each term by this factor, and then combining the results to express the polynomial in its factored form.

Highlights

The definition of the Greatest Common Monomial Factor (GCMF) in factoring polynomials.

The process of finding the GCMF involves identifying the highest common numerical coefficient and variable exponent among the terms of the polynomial.

Step-by-step instructions on factoring polynomials, starting with finding the GCMF.

Example given for factoring the polynomial 6x + 4x^2, demonstrating the steps to find and use the GCMF.

Explanation of how to determine the GCMF for numerical coefficients, using 6 and 4 as an example.

Determination of the GCMF for variables, identifying the least exponent among the variables present in the polynomial.

The GCMF for the polynomial 6x + 4x^2 is calculated as 2x.

Division of each term in the polynomial by the GCMF to simplify the expression.

Combining the simplified terms to express the factored form of the polynomial.

Introduction of a second example polynomial 9x^3y^7 - 18xy^2 - 36y.

Determination of the GCMF for the second example, focusing on the numerical and variable components separately.

Identification of the GCMF for the variables in the second example as y^2.

Calculation of the GCMF for the numerical part of the second example as 9.

Final GCMF for the second example polynomial is 9y^2.

Division of each term in the second polynomial by the GCMF to simplify.

Combining the simplified terms to present the factored form of the second polynomial.

Final factored form of the second polynomial is given as 9y^2(x^3y^5 - 2x - 4).

Transcripts

target is about

factor polynomials with greatest common

monomial

g c m

what is g c and f or the greatest common

denominator

it is the highest possible value of a

number

available or a combination of numbers

and variables which is common in each

term

of

let's say we have a x plus b x plus c

x the factors are

x times a plus b

plus c

in steps in factoring polynomials we do

see an

error first

find the greatest common monomial factor

second divide each term in the

polynomials

at x g c and f and the third

combine the answers in steps one

for the polynomial six x plus four x

squared

steps find the greatest common monomial

factor

in finding the agency and f first

determine the

biggest common factor of the numerical

coefficient

the numerical coefficient of

the terms are

six and four

the greatest common factor of 6

and 4 is 2

next determine the gcf of the variable

the common variable is

x to the first and then

x squared

the gcf of the variable is with the

least exponent

the least exponent between

x to the first and x squared is

one

so the jcf of the variable is

in the least exponent which is x at the

first

so we have x

therefore the greatest common monomial

factor

is 2

and then x which is 2x

divide each in the polynomial by

its latest formal

so our greatest common monomial factor

is 2x

divide 2x in each term of polynomial

6x plus 4x squared

so 6x divided by 2x

and 4x squared divided by 2x

so to simplify 6x divided by 2x is 3

and then 4x squared divided by 2x

is 2x and then combine the answers in

step 1

and 2 as equator so therefore

the factors of 6x plus 4x squared

are two x times

three plus two x

have one more example factor 9

x cube y to the 7

minus 18 x y squared minus 36

y to the

first we have to determine the gcf of

all the numerical partition

nine negative eighteen and negative

twenty six

the greatest common factor is

[Music]

then determining the gcf of the

variables

x to the third y to the seven x y

squared and then y to the third

since we don't have hormone

variable x

from the three terms so we don't have

any

uh common factor of x

but in y we have y

as the common variable for the three

terms

so we are going to get the less exponent

of

y which is two

so the gcf of the variable is y

squared

from the numeric column we appreciate

the gcf s9

and from the carnival the gcf

is twice here therefore the greatest

common monomial factor is

9y squared so we're going to divide

9y squared in each term

in the given polynomial so we have 9x

cube y to the seven divided by nine y

squared minus e to the x y squared

divided by nine y squared

minus five to six y to the third divided

by

nine y squared so we have x to the third

y to the eighth minus

two x minus four five

so therefore the factors of

the given polynomial are

nine y squared times

x to the third y to the p minus two x

minus four one