Rational Roots Reference Sheet
Factoring Blueprint
Mastering Polynomial Deconstruction
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1. Greatest Common Factor (GCF)
When to Use?
Check this FIRST! Use when every term shares a common factor.
General Form
\[ ab + ac = a(b + c) \]
Strategy
- Find largest number coefficient & smallest exponent.
- Divide terms by GCF and write leftovers in \(( )\).
Ex: \( 6x^3 + 12x^2 = 6x^2(x + 2) \)
2. Trinomials (\( a = 1 \))
When to Use?
Use for 3 terms in the form \( x^2 + bx + c \).
The "X" Method
Multiply \( c \)
Add \( b \)
Strategy
- Find 2 numbers that multiply to \( c \) and add to \( b \).
- Result: \( (x + p)(x + q) \)
Ex: \( x^2 + 5x + 6 = (x + 2)(x + 3) \)
3. Difference of Squares
When to Use?
2 perfect square terms with a minus sign.
General Form
\[ a^2 - b^2 = (a - b)(a + b) \]
Strategy
- Find root of 1st (\( a \)) and 2nd (\( b \)) terms.
- Two \(( )\): one with \( - \), one with \( + \).
Ex: \( x^2 - 16 = (x - 4)(x + 4) \)
The Golden Rule
Always check for a GCF first. After you factor once, check if the remaining parts can be factored again! Complete factoring means no more rules apply.
Rational Operations Lab
Working with Fractions of Functions
Multiplying & Dividing Factor & Simplify
Step-by-Step
- 1 Factor everything (top and bottom).
- 2 Divide? "Keep-Change-Flip" the 2nd fraction.
- 3 Simplify: Cancel common factors in top/bottom.
Example: Division
\[ \frac{x^2-4}{x} \div \frac{x+2}{x^2} \]
\[ \frac{(x-2)(x+2)}{x} \cdot \frac{x^2}{x+2} = x(x-2) \]
Adding & Subtracting Find the LCD
Step-by-Step
- 1 Factor denominators to find the LCD.
- 2 Multiply top/bottom by "missing" factors.
- 3 Combine numerators over the LCD & simplify.
Example: Subtraction
\[ \frac{5}{x} - \frac{2}{x+1} \]
\[ \frac{5(x+1) - 2x}{x(x+1)} = \frac{3x+5}{x(x+1)} \]
COMMON TRAP!
When subtracting, distribute the negative sign to the entire second numerator. Example: \( -(x + 3) \) becomes \( -x - 3 \).
Remember to state excluded values (where denominator = 0)!
