Chapter 2: Factoring Basic Expressions
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
To factor a basic algebraic expression, you must identify the GCF shared by both the numerical Coefficients and the Variables. The GCF is written outside the bracket, and the terms inside are found by dividing each original term by the GCF. This skill transforms an expression (a sum of terms) into a factored form (a product of factors), revealing the underlying structure and simplifying the expression for further analysis.
Audio Book
Dive deep into the subject with an immersive audiobook experience.
Factoring: The Reverse of Expansion
Chapter 1 of 1
π Unlock Audio Chapter
Sign up and enroll to access the full audio experience
Chapter Content
Factoring is the inverse of expanding brackets. It means rewriting an expression as a product of its factors by finding the Greatest Common Factor (GCF) shared by all the terms.
Detailed Explanation
Every term in an expression might share a common number or a common variable. The GCF is the largest number and the lowest power of the variable that they all share. We write this GCF outside the bracket and find the remaining terms by dividing the original terms by the GCF. This action reorganizes the expression from a sum to a product, which is often its most useful form.
Examples & Analogies
Imagine you have two boxes of cookies: one has 6 chocolate chip cookies ($6x$) and the other has 9 oatmeal cookies (9). You realize you bought them from the same baker (the GCF is 3). You can factor it as $3(2x + 3)$. This reveals that you bought 3 groups of cookies, where each group contained a certain mix.
\--
- Chunk Title: Finding the Greatest Common Factor (GCF)
- Chunk Text: To find the GCF of a multivariable expression, find the GCF of the Coefficients separately from the GCF of the Variables. The GCF for variables is always the lowest power of the shared variable.
- Detailed Explanation: If you have $4x^2 + 10x$, the coefficients (4 and 10) share a GCF of 2. The variables ($x^2$ and $x$) share a GCF of $x$. Therefore, the overall GCF is $2x$. When you divide $4x^2$ by $2x$, you are left with $2x$. When you divide $10x$ by $2x$, you are left with 5. This results in the factored form: $2x(2x + 5)$.
- Real-Life Example or Analogy: If you have four apples and six bananas ($4a + 6b$), the GCF is 2. The expression factors to $2(2a + 3b)$. This shows that the original terms are groups of 2.
Key Concepts
-
GCF: The core element to identify for factoring.
-
Inverse Operation: Factoring is the inverse of the distributive property (expanding).
-
Structure: Factoring reveals the components of an algebraic model.
-
-
Examples
-
Factoring $9ab - 15a^2b$:
-
GCF of 9 and 15 is 3.
-
GCF of $a$ and $a^2$ is $a$.
-
GCF of $b$ and $b$ is $b$.
-
Overall GCF is $3ab$. Result: $3ab(3 - 5a)$.
-
-
Flashcards
-
Term: What is the GCF of $12x^2$ and $18x$?
-
Definition: $6x$.
-
Term: Factoring is the reverse of what operation?
-
Definition: Expanding (Distributive Property).
-
Term: What must you do to check your factoring result?
-
Definition: Expand the factored expression; it must match the original expression.
-
-
Memory Aids
-
Rhyme: Look for the factor that terms want to share, pull it outside, and divide with care\!
-
Mnemonic: G.C.F.: Group Common Factors.
-
Visual Aid: Think of a reverse arrow going from the expanded form $ab + ac$ back to the factored form $a(b + c)$.
Examples & Applications
Factoring $9ab - 15a^2b$:
GCF of 9 and 15 is 3.
GCF of $a$ and $a^2$ is $a$.
GCF of $b$ and $b$ is $b$.
Overall GCF is $3ab$. Result: $3ab(3 - 5a)$.
Flashcards
Term: What is the GCF of $12x^2$ and $18x$?
Definition: $6x$.
Term: Factoring is the reverse of what operation?
Definition: Expanding (Distributive Property).
Term: What must you do to check your factoring result?
Definition: Expand the factored expression; it must match the original expression.
Memory Aids
Rhyme: Look for the factor that terms want to share, pull it outside, and divide with care\!
Mnemonic: G.C.F.: Group Common Factors.
Visual Aid: Think of a reverse arrow going from the expanded form $ab + ac$ back to the factored form $a(b + c)$.
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
Look for the factor that terms want to share, pull it outside, and divide with care\!
* **Mnemonic
Memory Tools
Group Common Factors.
* **Visual Aid
Flash Cards
Glossary
- Factoring
The process of rewriting an expression as a product of its GCF and the remaining terms in a bracket.
- Structure
Factoring reveals the components of an algebraic model.
- Definition
Expand the factored expression; it must match the original expression.
- Visual Aid
Think of a reverse arrow going from the expanded form $ab + ac$ back to the factored form $a(b + c)$.