Introduction to Factoring and Common Monomial Factors (GCMF)

TL;DR

Factoring is like reverse multiplication, breaking an expression into simpler parts (factors). We'll start by finding the Greatest Common Monomial Factor (GCMF), which is the biggest term that divides into every part of an expression. Once you find the GCMF, you can rewrite the original expression as the GCMF multiplied by what's left over.

1. The Mental Model

Think of factoring as "undoing" the distributive property. Instead of multiplying a term into parentheses, you're taking a common term out of an expression. It helps you simplify expressions and solve equations later on.

2. The Core Material

Factoring is essentially breaking down a polynomial into a product of simpler polynomials. When we talk about finding the Common Monomial Factor, we're looking for a single term (a monomial) that is a factor of every term in a given polynomial. The Greatest Common Monomial Factor (GCMF) is the largest of these common monomial factors.

Finding the GCMF

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To find the GCMF, you need to look at both the numerical coefficients and the variables:

1. Find the Greatest Common Factor (GCF) of the numerical coefficients: This is the largest number that divides evenly into all the coefficients.
2. Find the GCF of the variables: For each variable that appears in every term, take the variable with the smallest exponent. If a variable doesn't appear in every term, it's not part of the GCMF.
3. Multiply these results together: The product is your GCMF.

Factoring Out the GCMF

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Once you've found the GCMF, you factor it out by doing these steps:

1. Identify the GCMF of all terms in the polynomial.
2. Divide each term in the polynomial by the GCMF.
3. Write the GCMF outside parentheses, and inside the parentheses, write the results of your division.

Let's look at the process:

graph TD
    A["Start with the polynomial"] --> B{"Identify Numerical Coefficients"};
    B --> C{"Find GCF of Coefficients"};
    C --> D{"Identify Variables Present in ALL terms"};
    D --> E{"For each common variable, take the lowest exponent"};
    E --> F["Multiply (GCF of Coefficients) and (Common Variables with Lowest Exponents)"];
    F --> G["This is your GCMF"];
    G --> H{"Divide EACH term of original polynomial by the GCMF"};
    H --> I["Write: GCMF ( Results of division )"];
    I --> J["Factored Polynomial"];

Example: Finding GCMF

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Let's find the GCMF of $12x^3 - 18x^2 + 6x$.

1. Coefficients: The coefficients are 12, -18, and 6. The GCF of (12, 18, 6) is 6.
2. Variables: The variable 'x' appears in all terms. The exponents are $x^3$, $x^2$, and $x^1$. The smallest exponent is 1, so we take $x^1$ (or just x).
3. GCMF: Multiply the numerical GCF and the variable part: $6 * x = 6x$.

So, the GCMF is $6x$.

3. Worked Example

Factor the polynomial $10a^4b^2 - 15a^3b^3 + 5a^2b^2$.

1. Find the GCMF:

   • Coefficients: 10, -15, 5. The GCF of (10, 15, 5) is 5.
   • Variable 'a': $a^4$, $a^3$, $a^2$. The lowest exponent is $a^2$.
   • Variable 'b': $b^2$, $b^3$, $b^2$. The lowest exponent is $b^2$.
   • GCMF: $5a^2b^2$.

2. Divide each term by the GCMF:

   • Term 1: $\frac{10a^4b^2}{5a^2b^2} = 2a^{4-2}b^{2-2} = 2a^2b^0 = 2a^2$
   • Term 2: $\frac{-15a^3b^3}{5a^2b^2} = -3a^{3-2}b^{3-2} = -3ab$
   • Term 3: $\frac{5a^2b^2}{5a^2b^2} = 1a^{2-2}b^{2-2} = 1a^0b^0 = 1$

3. Write the factored form:
   $5a^2b^2(2a^2 - 3ab + 1)$

To check your answer, distribute $5a^2b^2$ back into the parentheses. You should get the original polynomial.

4. Key Takeaways

• Factoring is the reverse of multiplication, specifically the distributive property.
• The GCMF is the largest term (number, variable, or both) that divides evenly into every term of a polynomial.
• To find the GCMF, find the GCF of coefficients and the lowest power of common variables.
• Always write the GCMF outside the parentheses, followed by the remaining terms inside.
• You can check your factoring by multiplying the GCMF back into the parentheses.

Common Mistakes to Avoid:
* Forgetting to include a variable in the GCMF if it appears in every term.
* Including a variable in the GCMF if it doesn't appear in every term.
* Not taking the lowest exponent for common variables.
* Forgetting to write '1' as a placeholder when a whole term is divided out (like in the example, $5a^2b^2 / 5a^2b^2 = 1$).

5. Now Try It

Factor the polynomial $24x^5y^3 + 16x^3y^4 - 8x^2y^2$. Your answer should be in the form of GCMF(remaining polynomial). Success means you can multiply your factored form back out and get the original polynomial.