Factoring Techniques
Grade 10 · Math · Study Guide
Mastering the key methods for factoring polynomials: GCF, trinomials, difference of squares, and more.
Factoring is the process of rewriting a polynomial as a product of simpler polynomials. It's the algebraic equivalent of breaking a number into its prime factors. Every factoring problem should start with the same question: is there a Greatest Common Factor (GCF) I can pull out? The GCF is the largest factor that divides every term. Always factor it out first before trying any other technique.
Factoring trinomials of the form x² + bx + c means finding two numbers that multiply to c and add to b. These become the constants in two binomial factors: (x + p)(x + q) where p·q = c and p + q = b. For trinomials with a leading coefficient (ax² + bx + c), use the 'ac method': multiply a·c, find factors of that product that sum to b, split the middle term, then factor by grouping.
The difference of squares pattern is one of the most useful in algebra: a² - b² = (a + b)(a - b). Memorize this. It appears constantly in simplification and equation solving. Similarly, perfect square trinomials factor as a² + 2ab + b² = (a + b)² and a² - 2ab + b² = (a - b)².
For polynomials with four terms, try factoring by grouping: split into two pairs, factor each pair, then factor out the common binomial. Always verify your answer by multiplying the factors back out — if you get the original polynomial, you've factored correctly.
| Pattern Name | Form | Factored Form |
|---|---|---|
| Difference of Squares | a² - b² | (a + b)(a - b) |
| Perfect Square (sum) | a² + 2ab + b² | (a + b)² |
| Perfect Square (diff) | a² - 2ab + b² | (a - b)² |
| Sum of Cubes | a³ + b³ | (a + b)(a² - ab + b²) |
| Difference of Cubes | a³ - b³ | (a - b)(a² + ab + b²) |
Key terms
- GCF (Greatest Common Factor)
- The largest factor common to every term in the polynomial; always factor this out first.
- Factoring by Grouping
- Splitting a 4-term polynomial into two pairs, factoring each, then factoring the common binomial.
- Difference of Squares
- a² - b² = (a + b)(a - b); applies only to subtraction, not addition.
- Perfect Square Trinomial
- a² ± 2ab + b² = (a ± b)²; recognizable by the pattern of its terms.
- Prime Polynomial
- A polynomial that cannot be factored over the integers; analogous to a prime number.
Flashcards
- First step in factoring
- Always check for and factor out the GCF first
- Difference of squares
- a² - b² = (a + b)(a - b)
- Factoring x² + bx + c
- Find two numbers that multiply to c and add to b
- Perfect square trinomial
- a² ± 2ab + b² = (a ± b)²
- Factor by grouping
- For 4-term polynomials: group in pairs, factor each, pull out common binomial
- How to check factoring
- Multiply the factors back out — you should recover the original polynomial
Practice questions
Factor completely: x² - 16
- (x - 4)²
- (x - 4)(x + 4)
- (x + 4)²
- (x - 16)(x + 1)
Show answer
Answer: (x - 4)(x + 4)
x²-16 = x²-4² which fits the difference of squares pattern: (x-4)(x+4).
The GCF of 6x³ + 9x² - 3x is ___.
Show answer
Answer: 3x
The GCF of 6x³, 9x², and 3x is 3x. Factor: 3x(2x² + 3x - 1).
x² + 9 can be factored using the difference of squares pattern.
Show answer
Answer: False
The difference of squares requires subtraction: a²-b². The expression x²+9 is a sum of squares, which is prime over the reals.
Factor x² + 7x + 12:
- (x + 3)(x + 4)
- (x + 2)(x + 6)
- (x + 1)(x + 12)
- (x - 3)(x - 4)
Show answer
Answer: (x + 3)(x + 4)
Need two numbers that multiply to 12 and add to 7: 3 × 4 = 12, 3 + 4 = 7. So (x+3)(x+4).
x² - 10x + 25 is a perfect square trinomial.
Show answer
Answer: True
x²-10x+25 = x²-2(5)x+5² = (x-5)². The middle term matches -2ab with a=x and b=5.
Fun facts
- Factoring polynomials and breaking integers into prime factors are deeply related — in abstract algebra, polynomials form what's called a 'unique factorization domain,' just like the integers.
- The quadratic formula is essentially derived by factoring using completing the square — so every time you use it, you're really doing a disguised factoring operation.
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