x² + 7x + 12
The polynomial is written as a sum of three terms.
Free Algebra Guide
Factoring Polynomials
Learn how to factor out a greatest common factor, factor monic and non-monic trinomials, recognize a difference of squares, use the AC method, and check your factors by multiplying.
Factoring rewrites a polynomial as a product of simpler expressions. It reverses multiplication: instead of multiplying factors to create a polynomial, you begin with the polynomial and determine which factors produced it. A reliable process starts by checking for a greatest common factor and then identifying the polynomial pattern.
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What Does It Mean to Factor a Polynomial?
A polynomial is factored when it is written as a product. For example, multiplying the binomials (x + 3) and (x + 4) produces x² + 7x + 12. Factoring starts with x² + 7x + 12 and reverses that multiplication.
(x + 3)(x + 4)
The same polynomial is written as a product of two binomials.
(x + 3)(x + 4) = x² + 7x + 12
Check for a Greatest Common Factor First
Before using any other factoring method, determine whether every term shares a numerical factor, a variable factor, or both. Use the greatest common divisor of the coefficients and the smallest exponent shared by each variable.
Example: 12x³ + 18x²
Coefficient GCF
GCF(12, 18) = 6
Shared variable
The smallest shared power is x².
Complete GCF
6x²
Factor it out
12x³ + 18x² = 6x²(2x + 3)
Do Not Stop Too Early
Factoring out 3x² is possible, but it is not the greatest common factor. Use 6x² so the expression is factored completely.
Factoring Monic Trinomials
A monic quadratic trinomial has the form x² + bx + c. Find two integers whose product is c and whose sum is b. Those integers become the constant terms in the binomial factors.
Example: x² + 7x + 12
Required product
12
Required sum
7
Number pair
3 and 4, because 3 · 4 = 12 and 3 + 4 = 7
Factored form
(x + 3)(x + 4)
How to Choose the Correct Signs
| Sign of c | Sign of b | Factor signs | Example |
|---|---|---|---|
| Positive | Positive | Both positive | x² + 7x + 12 = (x + 3)(x + 4) |
| Positive | Negative | Both negative | x² − 7x + 12 = (x − 3)(x − 4) |
| Negative | Either | One positive and one negative | x² − x − 12 = (x − 4)(x + 3) |
The Order Does Not Matter
Because multiplication is commutative, (x + 3)(x + 4) and (x + 4)(x + 3) are equivalent.
Factoring a Difference of Squares
A difference of squares has two perfect-square terms separated by subtraction.
a² − b² = (a − b)(a + b)
Example: 9x² − 25
First square root
√(9x²) = 3x
Second square root
√25 = 5
Conjugate factors
(3x − 5)(3x + 5)
It Must Be a Difference
The expression a² + b² does not factor over the real numbers using the difference-of-squares pattern.
Factoring Non-Monic Trinomials
A non-monic quadratic trinomial has the form ax² + bx + c with a ≠ 1. The AC method converts the trinomial into four terms so it can be factored by grouping.
The Central Idea
Find two integers whose product is ac and whose sum is b. Use them to split only the middle term. The original constant c stays unchanged.
Using the AC Method
Example: 5x² + 12x + 7
Identify a, b, and c
a = 5, b = 12, c = 7
Calculate ac
5 · 7 = 35
Find the pair
5 and 7 multiply to 35 and add to 12.
Split the middle term
5x² + 5x + 7x + 7
Do Not Replace c With ac
The number 35 helps you find the middle-term pair, but the last term remains 7. The correct rewrite ends in + 7, not + 35.
Either Middle-Term Order Can Work
You may write 5x² + 5x + 7x + 7 or 5x² + 7x + 5x + 7. Choose the order that creates convenient groups.
Factor the Four Terms by Grouping
Continue: 5x² + 5x + 7x + 7
Create two groups
(5x² + 5x) + (7x + 7)
Factor each group
5x(x + 1) + 7(x + 1)
Shared binomial
Both groups contain (x + 1).
Final factors
(5x + 7)(x + 1)
What Does It Mean to Factor Completely?
A polynomial is factored completely only when no remaining factor can be factored further using the available number system.
Example: 2x² − 18
Factor the GCF
2(x² − 9)
Recognize the pattern
x² − 9 is a difference of squares.
Factor completely
2(x − 3)(x + 3)
Check Your Factors by Multiplying
Multiply the factors using distribution or the box method. If the expanded result matches the original polynomial exactly, the factorization is correct.
Verify: (5x + 7)(x + 1)
Multiply
5x² + 5x + 7x + 7
Combine like terms
5x² + 12x + 7
Compare
The result matches the original polynomial.
Worked Factoring Examples
15x² + 20x
- The coefficient GCF is 5.
- Both terms contain x.
- Result: 5x(3x + 4).
x² − 8x + 15
- Product 15; sum −8.
- The pair is −3 and −5.
- Result: (x − 3)(x − 5).
x² + 2x − 15
- Product −15; sum 2.
- The pair is 5 and −3.
- Result: (x + 5)(x − 3).
16x² − 81
- The square roots are 4x and 9.
- Use conjugate signs.
- Result: (4x − 9)(4x + 9).
6x² + 11x + 3
- ac = 18; the pair is 9 and 2.
- Rewrite as 6x² + 9x + 2x + 3.
- Result: (3x + 1)(2x + 3).
3x² − 27
- Factor out 3: 3(x² − 9).
- Factor the difference of squares.
- Result: 3(x − 3)(x + 3).
Common Mistakes When Factoring Polynomials
Skipping the GCF
Always check for a shared factor before using another method.
Using Only the Product
The number pair must satisfy both the required product and the required sum.
Choosing the Wrong Signs
Use the sign of the product and the sign of the sum together.
Replacing c With ac
The value ac helps split the middle term; it does not replace the original constant.
Stopping After Splitting the Middle Term
After creating four terms, factor by grouping and then factor out the shared binomial.
Not Factoring Completely
Check every remaining factor for a GCF, trinomial pattern, or difference of squares.
Using Difference of Squares on a Sum
The pattern a² − b² requires subtraction, not addition.
Not Checking the Result
Multiply the factors to confirm that every original coefficient and sign returns.
A Reliable Strategy for Factoring Polynomials
1. Arrange the terms
Write the polynomial in descending powers of x.
2. Check for a GCF
Factor out the greatest number and smallest shared variable power.
3. Count the terms
Two terms may form a difference of squares; three terms may form a quadratic trinomial.
4. Identify the pattern
Choose GCF, monic trinomial, difference of squares, or the AC method.
5. Build the factors
Use the required product and sum, or split and group when a ≠ 1.
6. Factor completely
Inspect every remaining factor before stopping.
7. Multiply to verify
The factors must reproduce the exact original polynomial.
Practice Factoring Polynomials
Enter every step yourself while practicing GCF factoring, monic trinomials, difference of squares, non-monic trinomials, the AC method, and grouping.
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