Factor Polynomials with GCMF Video
Summary
TLDRThis 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
📚 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.
🔍 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.
Mindmap
Keywords
💡Factor Polynomials
💡Greatest Common Monomial Factor (GCMF)
💡Greatest Common Factor (GCF)
💡Monomial
💡Variable
💡Least Exponent
💡Numerical Coefficient
💡Divide Each Term
💡Simplify
💡Polynomial
💡Steps in Factoring
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
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