Polynomials and Factoring

TL;DR

Polynomials are expressions made of variables and coefficients, combined using addition, subtraction, and multiplication, with non-negative integer exponents. Factoring a polynomial means breaking it down into a product of simpler polynomials, like finding the prime factors of a number. This skill simplifies expressions, helps solve equations, and is fundamental for higher-level math.

1. The Mental Model

Think of polynomials as math's version of building blocks. Factoring is like deconstructing a complex LEGO model back into its individual pieces. Just as you can rebuild the model from its pieces, you can multiply the factors back together to get the original polynomial.

2. The Core Material

Polynomials are expressions like $3x^2 + 2x - 5$. They're made of terms, where each term is a product of a number (coefficient) and one or more variables raised to a non-negative integer power.

Key Definitions:
* Term: A single part of a polynomial, like $3x^2$, $2x$, or $-5$.
* Coefficient: The numerical part of a term, like 3 in $3x^2$.
* Variable: The letter part, like $x$.
* Exponent (Degree of the term): The power to which the variable is raised, like 2 in $x^2$.
* Degree of a Polynomial: The highest exponent of the variable in the polynomial. For $3x^2 + 2x - 5$, the degree is 2.
* Constant Term: A term without a variable, like -5.

Types of Polynomials by Number of Terms:
* Monomial: One term (e.g., $5x^3$)
* Binomial: Two terms (e.g., $2x + 7$)
* Trinomial: Three terms (e.g., $x^2 - 3x + 1$)

Factoring Polynomials is the reverse of multiplying them. It's about finding expressions that, when multiplied together, give you the original polynomial.

2.1. Basic Factoring: Greatest Common Factor (GCF)

Always start by looking for a GCF! This is the largest term (number, variable, or both) that divides evenly into every term of the polynomial.

Example: Factor $6x^3 - 9x^2 + 12x$
1. Find the GCF of the coefficients (6, 9, 12): It's 3.
2. Find the GCF of the variable parts ($x^3, x^2, x$): It's $x$ (always the lowest power).
3. The overall GCF is $3x$.
4. Divide each term by the GCF:
* $6x^3 / (3x) = 2x^2$
* $-9x^2 / (3x) = -3x$
* $12x / (3x) = 4$
5. Write the GCF outside parentheses, with the results inside: $3x(2x^2 - 3x + 4)$

2.2. Factoring Binomials: Special Products

Some binomials follow specific patterns:

• Difference of Squares: $a^2 - b^2 = (a - b)(a + b)$
  • Example: $x^2 - 25 = (x - 5)(x + 5)$
  • Example: $4y^2 - 9 = (2y - 3)(2y + 3)$
• Sum/Difference of Cubes:
  • $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$
  • $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$
  • Example: $x^3 + 8 = (x + 2)(x^2 - 2x + 4)$

2.3. Factoring Trinomials ($ax^2 + bx + c$)

Case 1: When $a = 1$ (e.g., $x^2 + bx + c$)
Find two numbers that multiply to $c$ and add up to $b$.

Example: Factor $x^2 + 5x + 6$
1. We need two numbers that multiply to 6 and add to 5.
2. The numbers are 2 and 3.
3. So, the factored form is $(x + 2)(x + 3)$.

Case 2: When $a \neq 1$ (e.g., $2x^2 + 7x + 3$)
This is often done using the "AC method" or "grouping."

Example: Factor $2x^2 + 7x + 3$
1. Multiply $a$ and $c$: $2 \times 3 = 6$.
2. Find two numbers that multiply to 6 and add to $b$ (which is 7). These are 1 and 6.
3. Rewrite the middle term ($7x$) using these two numbers: $2x^2 + 1x + 6x + 3$.
4. Group the terms: $(2x^2 + 1x) + (6x + 3)$.
5. Factor out the GCF from each group: $x(2x + 1) + 3(2x + 1)$.
6. Notice the common binomial factor $(2x + 1)$. Factor it out: $(2x + 1)(x + 3)$.

2.4. Factoring by Grouping (for 4 terms)

This is essentially what we did in Case 2 for trinomials with $a \neq 1$. You can use it whenever you have four terms.

Example: Factor $x^3 + 2x^2 + 5x + 10$
1. Group the terms: $(x^3 + 2x^2) + (5x + 10)$.
2. Factor out the GCF from each group: $x^2(x + 2) + 5(x + 2)$.
3. Factor out the common binomial $(x + 2)$: $(x + 2)(x^2 + 5)$.

Here's a flowchart to help you decide which factoring method to use:

graph TD
    A["Start Factoring"] --> B{Is there a GCF for ALL terms?};
    B -- Yes --> C[Factor out GCF];
    C --> D{How many terms are left inside the parentheses?};
    B -- No --> D;
    D -- 2 Terms --> E{Is it a Difference of Squares (a² - b²)?};
    E -- Yes --> F[Factor as (a - b)(a + b)];
    E -- No --> G{Is it Sum/Difference of Cubes (a³ ± b³)?};
    G -- Yes --> H[Factor using cube formulas];
    G -- No --> I["Cannot factor this binomial (usually)"];
    D -- 3 Terms --> J{Is a=1 (x² + bx + c)?};
    J -- Yes --> K[Find two numbers that mult to c, add to b];
    J -- No (a≠1) --> L[Use AC Method (grouping)];
    D -- 4 Terms --> M[Factor by Grouping];
    F --> N[Done];
    H --> N;
    K --> N;
    L --> N;
    M --> N;
    I --> N;

3. Worked Example

Let's factor the polynomial $3x^3 - 12x^2 - 15x$.

1. Look for a GCF:
   • Coefficients: 3, -12, -15. The GCF is 3.
   • Variables: $x^3, x^2, x$. The GCF is $x$.
   • Overall GCF: $3x$.
2. Factor out the GCF:
   $3x(x^2 - 4x - 5)$
3. Now, look at the trinomial inside the parentheses: $x^2 - 4x - 5$.
   • This is a trinomial where $a = 1$.
   • We need two numbers that multiply to $c$ (-5) and add to $b$ (-4).
   • The numbers are -5 and 1 (since $(-5) \times 1 = -5$ and $-5 + 1 = -4$).
4. Factor the trinomial:
   $(x - 5)(x + 1)$
5. Combine the GCF with the factored trinomial:
   The fully factored polynomial is $3x(x - 5)(x + 1)$.

To check your work, you can multiply the factors back together to ensure you get the original polynomial.

4. Key Takeaways

• Always check for a Greatest Common Factor (GCF) first, no matter how many terms.
• Know the special product formulas for difference of squares and sum/difference of cubes.
• For trinomials ($ax^2 + bx + c$), remember the simple "multiply to $c$, add to $b$" rule when $a=1$.
• When $a \neq 1$ in a trinomial, use the AC method (or grouping) to factor it.
• Factoring by grouping is your go-to for polynomials with four terms.
• Factoring helps simplify expressions and is crucial for solving polynomial equations.

Common Mistakes to Avoid:
- Forgetting to look for a GCF before trying other factoring methods.
- Incorrectly applying the sum/difference of squares formula to a sum of squares (e.g., $a^2 + b^2$ usually doesn't factor over real numbers).
- Making sign errors when finding numbers that multiply to $c$ and add to $b$.
- Not fully factoring the polynomial; always check if any of your factors can be factored further.

5. Now Try It

Factor the polynomial $2x^4 - 50x^2$. Spend about 15 minutes on it.

• What to do:
  1. Start by finding the GCF for all terms.
  2. Factor out the GCF.
  3. Look at the remaining expression inside the parentheses. What type of polynomial is it? Can it be factored further using any of the methods we discussed?
  4. Continue factoring until no part of the polynomial can be broken down any further.
• What success looks like: You'll end up with three factors multiplied together. The final answer should be $2x^2(x - 5)(x + 5)$.