🎯 Learning Objectives
- What a polynomial is and how to identify its degree.
- The four go-to factoring techniques used on the SAT.
- How to spot the difference of squares pattern.
- How to simplify rational expressions by canceling common factors.
- Strategies for FOIL and reverse-FOIL.
- Practice questions with worked solutions.
1. Polynomial Vocabulary
Term | Meaning | Example |
Polynomial | A sum of terms with whole-number powers of x | 3x² + 5x − 7 |
Degree | The highest exponent of x | Degree 2 (quadratic) |
Coefficient | Number multiplying the variable | 3 in 3x² |
Constant term | Term with no variable | −7 |
Like terms | Same variable AND same power | 3x² and −5x² |
2. The 4 Essential Factoring Techniques
Technique 1: Greatest Common Factor (GCF)
Always check this first. Pull out the largest factor common to every term.
GCF example
6x² + 9x = 3x(2x + 3)
Technique 2: Difference of Squares
Memorize this!
a² − b² = (a − b)(a + b)
Recognize: ANY two perfect squares with a minus sign between them. Examples: x² − 25 = (x−5)(x+5), 9x² − 16 = (3x−4)(3x+4).
Technique 3: Trinomial Factoring (x² + bx + c)
Find two numbers that MULTIPLY to c and ADD to b. Then write (x + p)(x + q).
📝 Example: Factor x² + 7x + 12
Need: Two numbers that multiply to 12 and add to 7.
Try: 3 and 4. (3 × 4 = 12 ✓, 3 + 4 = 7 ✓)
Result: (x + 3)(x + 4)
Technique 4: Perfect Square Trinomial
and a² − 2ab + b² = (a − b)²
a² + 2ab + b² = (a + b)²
Spot the pattern: first and last terms are perfect squares, middle term is twice their product.
3. The Factoring Decision Flowchart
STEP 1: Always pull out a GCF first if there is one
STEP 2: Two terms? Check for difference of squares
STEP 3: Three terms? Try trinomial factoring
STEP 4: Four+ terms? Try grouping
4. Simplifying Rational Expressions
A rational expression is just a polynomial fraction. Simplify by factoring the top and bottom, then canceling common factors.
📝 Example: Simplify (x² − 9) / (x² − 6x + 9)
Top: x² − 9 = (x − 3)(x + 3)
Bottom: x² − 6x + 9 = (x − 3)(x − 3)
Cancel: (x − 3)(x + 3) / (x − 3)(x − 3) = (x + 3)/(x − 3)
5. Strategies
⚡ STRATEGY 1: Difference of squares is everywhere
If you see TWO TERMS connected by a minus sign, immediately check if both are perfect squares. Verify that both terms are squares before using the identity.
⚡ STRATEGY 2: FOIL backward for trinomials
x² + 7x + 12 → look for two numbers (3, 4) that multiply to 12 and add to 7. Then read off (x+3)(x+4).
⚡ STRATEGY 3: When in doubt, FOIL the answer choices
If you can't factor, expand each multiple choice option using FOIL and see which matches the original.
⚠️ Common Mistakes
- Forgetting to extract the GCF first: it makes everything else easier.
- Sign errors: x² − 9 = (x−3)(x+3), NOT (x−3)(x−3).
- Canceling terms instead of factors. You can only cancel WHOLE multiplicative factors, not single terms in a sum.
- Stopping after one technique when more factoring is possible.
6. Summary
📌 Key Takeaways
- Always: Pull out GCF first.
- Two terms minus: Difference of squares: a² − b² = (a−b)(a+b).
- Three terms: Find numbers that multiply to c and add to b.
- Perfect square: a² ± 2ab + b² = (a ± b)²
- Rational expression: Factor top and bottom, then cancel common factors.
7. Practice: 10 Questions
Q1. Factor: x² − 16
A) (x−4)² B) (x−4)(x+4) C) (x+4)² D) (x−16)(x+1)
Hint: Difference of squares.
Q2. Factor: x² + 8x + 15
A) (x+3)(x+5) B) (x+1)(x+15) C) (x−3)(x−5) D) (x+5)(x+3)
Hint: 3 and 5: multiply to 15, add to 8.
Q3. Factor: 4x² − 9
A) (2x−3)(2x+3) B) (4x−3)(x+3) C) (2x−9)(2x+1) D) (4x−9)(x+1)
Hint: Difference of squares: (2x)² − 3².
Q4. Factor completely: 3x² − 12
A) 3(x²−4) B) 3(x−2)(x+2) C) (3x−6)(x+2) D) (x−2)(3x+6)
Hint: GCF 3 first, then difference of squares.
Q5. Simplify: (x² − 4)/(x − 2)
A) x + 2 B) x − 2 C) x² D) x
Hint: Top factors as (x−2)(x+2). Cancel (x−2).
Q6. Factor: x² − 6x + 9
A) (x−3)² B) (x+3)² C) (x−3)(x+3) D) (x−9)(x+1)
Hint: Perfect square trinomial.
Q7. Factor: x² − x − 12
A) (x−4)(x+3) B) (x+4)(x−3) C) (x−12)(x+1) D) (x−6)(x+2)
Hint: Multiply to −12, add to −1: −4 and 3.
Q8. Simplify: (x² + 5x + 6)/(x + 2)
A) x + 3 B) x + 2 C) x − 3 D) x
Hint: (x+2)(x+3)/(x+2) = x + 3.
Q9. Factor: 2x² + 7x + 3
A) (2x+1)(x+3) B) (2x+3)(x+1) C) (x+3)(2x−1) D) (x+1)(2x−3)
Hint: FOIL each: (2x+1)(x+3) gives 2x²+7x+3 ✓
Q10. Simplify: (x² − 9)/(x² + 6x + 9)
A) (x−3)/(x+3) B) (x+3)/(x−3) C) x−3 D) 1
Hint: Top: (x−3)(x+3). Bottom: (x+3)². Cancel (x+3).
Answer Key & Worked Solutions
# | Answer | Reasoning |
Q1. | B) (x−4)(x+4) | Difference of squares. |
Q2. | A) (x+3)(x+5) | 3 + 5 = 8, 3 × 5 = 15. |
Q3. | A) (2x−3)(2x+3) | (2x)² − 3². |
Q4. | B) 3(x−2)(x+2) | GCF then difference of squares. |
Q5. | A) x + 2 | Cancel (x − 2). |
Q6. | A) (x−3)² | Perfect square: a² − 2ab + b². |
Q7. | A) (x−4)(x+3) | −4 + 3 = −1, −4 × 3 = −12. |
Q8. | A) x + 3 | Cancel (x + 2). |
Q9. | A) (2x+1)(x+3) | Verify by FOIL. |
Q10. | A) (x−3)/(x+3) | Cancel (x+3) once. |