What Does It Mean to Factor a Polynomial?
Factoring a polynomial means rewriting it as a product of simpler polynomials. Just as 12 can be factored into 2 x 2 x 3, the polynomial x^2 + 5x + 6 can be factored into (x + 2)(x + 3). Factoring is the reverse of multiplying — instead of expanding (x + 2)(x + 3) into x^2 + 5x + 6, you start with x^2 + 5x + 6 and find the factors that produce it.
Factoring is one of the most important skills in algebra because it is used everywhere: solving equations, simplifying rational expressions, finding zeros of functions, graphing parabolas, and integrating in calculus. If you can factor fluently, many problems that look complicated become straightforward.
There are several factoring methods, and the key to success is knowing which method to apply for each type of polynomial. This guide covers all the major methods in the order you should try them: GCF first, then special patterns, then trinomial factoring, and finally grouping.
Step 1: Always Factor Out the GCF First
The greatest common factor (GCF) is the largest expression that divides evenly into every term of the polynomial. Always look for a GCF before trying any other method. Factoring out the GCF simplifies the remaining polynomial and often reveals a pattern you would otherwise miss.
To find the GCF, identify the largest number that divides all coefficients and the lowest power of each variable that appears in every term. For 6x^3 + 9x^2 + 3x, the GCF is 3x (3 divides 6, 9, and 3; x is the lowest power of x present). Factoring gives 3x(2x^2 + 3x + 1).
For 12x^4y^2 - 8x^3y^3 + 4x^2y^2, the GCF is 4x^2y^2. Factoring gives 4x^2y^2(3x^2 - 2xy + 1). Always verify by distributing the GCF back through — you should get the original polynomial.
Do not skip the GCF step. Students who jump straight to trinomial factoring often struggle because they are trying to factor a polynomial that has a common factor making the numbers unnecessarily large. Factor out the GCF first, and the remaining polynomial becomes much easier to work with.
Difference of Squares: a^2 - b^2 = (a + b)(a - b)
The difference of squares pattern applies when a polynomial has exactly two terms, both are perfect squares, and they are subtracted. The formula is a^2 - b^2 = (a + b)(a - b). This pattern comes up constantly and is worth memorizing.
Examples: x^2 - 9 = (x + 3)(x - 3). 4x^2 - 25 = (2x + 5)(2x - 5). 49a^2 - 16b^2 = (7a + 4b)(7a - 4b). In each case, identify what is being squared in each term, then write the sum and difference of those square roots.
Important: the SUM of squares (a^2 + b^2) does not factor over the real numbers. If you see x^2 + 9, it is already fully factored (it cannot be broken down further with real numbers). This is a common mistake — students try to factor sums of squares and write incorrect factors.
Watch for nested differences of squares. The expression x^4 - 16 is a difference of squares: (x^2)^2 - 4^2 = (x^2 + 4)(x^2 - 4). But x^2 - 4 is itself a difference of squares: (x + 2)(x - 2). So the complete factorization is (x^2 + 4)(x + 2)(x - 2). Always check whether the resulting factors can be factored further.
Perfect Square Trinomials
A perfect square trinomial is a trinomial that results from squaring a binomial. There are two patterns: a^2 + 2ab + b^2 = (a + b)^2 and a^2 - 2ab + b^2 = (a - b)^2. Recognizing these saves time because you can factor them instantly.
How to identify a perfect square trinomial: the first and last terms must be perfect squares, and the middle term must equal 2 times the product of the square roots of the first and last terms. For x^2 + 10x + 25: x^2 is a perfect square (x), 25 is a perfect square (5), and 10x = 2(x)(5). So x^2 + 10x + 25 = (x + 5)^2.
Another example: 9x^2 - 24x + 16. First term: 9x^2 = (3x)^2. Last term: 16 = 4^2. Middle term: 24x = 2(3x)(4). Check: it matches the pattern with a minus sign. So 9x^2 - 24x + 16 = (3x - 4)^2.
If the middle term does not match the 2ab pattern, the trinomial is not a perfect square. For x^2 + 8x + 25, check: 2(x)(5) = 10x, but the middle term is 8x. Not a perfect square trinomial — use the general trinomial method instead.
Factoring Trinomials: x^2 + bx + c
For trinomials with a leading coefficient of 1 (like x^2 + 7x + 12), find two numbers that multiply to c (12) and add to b (7). Those numbers are 3 and 4, because 3 x 4 = 12 and 3 + 4 = 7. The factored form is (x + 3)(x + 4).
The signs matter. For x^2 - 7x + 12, you need two numbers that multiply to +12 and add to -7. Both numbers must be negative: -3 and -4. Factor: (x - 3)(x - 4). For x^2 + x - 12, you need two numbers that multiply to -12 and add to +1. Those are +4 and -3. Factor: (x + 4)(x - 3).
Sign rules for x^2 + bx + c: If c is positive and b is positive, both numbers are positive. If c is positive and b is negative, both numbers are negative. If c is negative, one number is positive and one is negative (the larger one matches the sign of b).
When you cannot find integer factors, the trinomial might not factor over the integers. Use the discriminant test: b^2 - 4ac. If it is a perfect square, the trinomial factors with integers. If not, you would need the quadratic formula or irrational/complex factors.
Factoring Trinomials: ax^2 + bx + c (a is not 1)
When the leading coefficient is not 1, factoring is harder. The AC method (also called the product-sum method) is the most reliable approach. Multiply a and c to get the product ac. Find two numbers that multiply to ac and add to b. Use these numbers to split the middle term, then factor by grouping.
Example: Factor 6x^2 + 11x + 3. Multiply a x c: 6 x 3 = 18. Find two numbers that multiply to 18 and add to 11: those are 9 and 2. Rewrite the middle term: 6x^2 + 9x + 2x + 3. Group: (6x^2 + 9x) + (2x + 3). Factor each group: 3x(2x + 3) + 1(2x + 3). Factor out the common binomial: (3x + 1)(2x + 3).
Example: Factor 2x^2 - 7x + 3. Product: 2 x 3 = 6. Find two numbers that multiply to 6 and add to -7: those are -6 and -1. Rewrite: 2x^2 - 6x - x + 3. Group: (2x^2 - 6x) + (-x + 3). Factor: 2x(x - 3) - 1(x - 3). Result: (2x - 1)(x - 3).
Always verify your factoring by multiplying the factors back together using FOIL or distribution. If you do not get the original polynomial, recheck your work. This verification step catches errors and builds confidence. If you are struggling with factoring a specific polynomial, ScanSolve can show you the step-by-step solution.
Factoring by Grouping (Four or More Terms)
When a polynomial has four terms, factoring by grouping is often the right approach. Group the terms into pairs, factor each pair separately, then look for a common binomial factor.
Example: Factor x^3 + 2x^2 + 3x + 6. Group: (x^3 + 2x^2) + (3x + 6). Factor each group: x^2(x + 2) + 3(x + 2). Both groups contain (x + 2) as a factor: (x^2 + 3)(x + 2).
Sometimes the grouping is not obvious and you need to rearrange the terms. For x^3 + 3x + 2x^2 + 6, rearranging to x^3 + 2x^2 + 3x + 6 (collecting terms with shared factors) makes the grouping clear. If the first grouping does not produce a common factor, try different groupings.
Grouping also works with more complex expressions. For 2x^3 - 3x^2 - 8x + 12: Group as (2x^3 - 3x^2) + (-8x + 12). Factor: x^2(2x - 3) - 4(2x - 3). Common factor (2x - 3): (x^2 - 4)(2x - 3). But x^2 - 4 is a difference of squares: (x + 2)(x - 2)(2x - 3). Always check whether resulting factors can be factored further.
The Factor Theorem and Synthetic Division
The Factor Theorem states that (x - r) is a factor of a polynomial P(x) if and only if P(r) = 0. This means you can test potential factors by plugging values into the polynomial. If the result is zero, you have found a factor.
For higher-degree polynomials, the Rational Root Theorem helps narrow down which values to test. The possible rational roots are all fractions p/q where p divides the constant term and q divides the leading coefficient. For 2x^3 + 3x^2 - 8x + 3, the constant term is 3 (factors: 1, 3) and the leading coefficient is 2 (factors: 1, 2). Possible rational roots: plus or minus 1, 3, 1/2, 3/2.
Test x = 1: P(1) = 2 + 3 - 8 + 3 = 0. Since P(1) = 0, (x - 1) is a factor. Use synthetic division or polynomial long division to divide 2x^3 + 3x^2 - 8x + 3 by (x - 1). The result is 2x^2 + 5x - 3, which factors as (2x - 1)(x + 3). Complete factorization: (x - 1)(2x - 1)(x + 3).
Synthetic division is a shortcut for dividing by (x - r). Write only the coefficients, bring down the first coefficient, multiply by r, add to the next coefficient, and repeat. It is faster than polynomial long division and less prone to errors. For difficult polynomials, combining the Rational Root Theorem with synthetic division is the most efficient approach.
