Welcome to the beginning of polynomial factoring! In this lesson, we start breaking down polynomials into simpler components—a crucial skill for solving equations easily. To kick things off, we are focusing on the fundamental technique: removing the Greatest Common Factor (GCF).

The Concept: Reverse Distributive Property

Recall the Distributive Property, which states that $a(b + c) = ab + ac$. Factoring by GCF is simply doing this in reverse. We look for a term ($a$) that divides evenly into every part of the polynomial to write it in a factored form.

For example, looking at the polynomial $2x^2 - 4$, we can see that both terms share a factor of $2$:

$$2x^2 - 4 = 2(x^2 - 2)$$

Key Strategies from Class

  • Finding the GCF: Look for the largest number and the highest power of variables that every term shares. For example, in $8x^3 - 4x^2 - 16x$, the GCF is $4x$.
  • Handling Negatives: If the first term is negative, it is often helpful to factor out a negative GCF. For instance, with $-14x - 12x^2$, factor out $-2x$ to get $-2x(7 + 6x)$.
  • Factoring by Grouping: When you have a polynomial with four terms, such as $6h^4 - 4h^3 + 12h - 8$, you can group terms to find a common binomial factor.
    • Group terms: $(6h^4 - 4h^3) + (12h - 8)$
    • Factor GCF from each group: $2h^3(3h - 2) + 4(3h - 2)$
    • Factor out the common binomial: $(3h - 2)(2h^3 + 4)$

Real-World Application

Factoring is useful for geometry problems. If the area of a squash court is $9x^2 + 6x$ square meters, we can factor out $3x$ to find expressions for the dimensions: $3x(3x + 2)$. This means the possible width is $3x$ and the length is $3x + 2$.

Class Resources

Discussion Question: The "Best Friend" Factor

In this lesson, we look for things that terms have in common to pull them out, allowing us to separate the similarities from the differences. Let's apply this logic to real life!

Your Task: Think about you and your best friend.

  1. List three things you both have in common (your GCF).
  2. List one thing that is unique to you and one thing unique to your friend.
  3. "Factor" your friendship in the comments below using this format:

(Common Trait 1 + Common Trait 2 + Common Trait 3) [ (My Unique Trait) + (Friend's Unique Trait) ]