Factor completely means to write a polynomial as a product of factors that cannot be factored any further. First take out the greatest common factor, then factor what is left: 2x2 − 18 = 2(x2 − 9) = 2(x + 3)(x − 3). Stopping at 2(x2 − 9) is not complete, because x2 − 9 still factors.
Taking out the GCF is only step 1. Keep going until every factor is prime: here x2 − 4 still splits into (x + 2)(x − 2).
Formal definition and notation
A polynomial with integer coefficients is factored completely when it is written as a product of a constant (its greatest common factor) and polynomials that are each prime: none of them can be written as a product of two polynomials of lower degree with integer coefficients. For example, x2 + 4 is prime over the integers, so (x2 + 4)(x + 2)(x − 2) is the complete factorization of x4 − 16.
Key facts
Always start with the greatest common factor. It makes the remaining numbers smaller and easier to factor.
look for a difference of squares: a2 − b2 = (a + b)(a − b)
3 terms
factor the trinomial: x2 + bx + c = (x + m)(x + n) with mn = c, m + n = b
4 terms
try grouping: factor pairs of terms, then the common binomial
After each step, look at every factor again. You are done only when no factor can be split further. Multiply back to check.
Try it: the number version
Interactive
Factoring a number completely means splitting it until every branch is prime. Try 180, then the samples: a polynomial is factored completely in the same spirit, when no factor can be split again.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Factor Completely examples, step by step
Example 1: GCF only
Warm-up
Factor completely: 5x2 + 15x.
GCF of 5x2 and 15x is 5x.
Divide each term by 5x: 5x(x + 3).
x + 3 has degree 1, so nothing else factors.
Answer:5x2 + 15x = 5x(x + 3), which is factored completely.
factoring is any rewriting as a product, even a partial one like 2(x2 − 9). Factor completely asks for the last step, where no factor can be split further.
Factor Completelyvs.Simplify
Simplify usually means combine like terms or reduce, often giving a sum. Factor completely always ends in a product.
Factor Completelyvs.Solve
Solving x2 − 9 = 0 gives the values x = 3 and x = −3. Factoring completely only rewrites the expression: (x + 3)(x − 3).
Quick check: 3 questions
Question 1 of 3
Which is 6x2 − 24 factored completely?
Show answer and explanation
Answer: C. GCF 6 leaves x2 − 4 = (x + 2)(x − 2). A stops too early, B leaves a factor of 2 inside 6x + 12, and D multiplies to 6x2 − 24x + 24.
Question 2 of 3
Which polynomial is already factored completely?
Show answer and explanation
Answer: A.x2 + 1 is a sum of squares and does not factor over the integers. B and D contain differences of squares, and C still has a common factor of 2 in 2x + 6.
Question 3 of 3
Factor completely: 2x2 + 14x + 24.
Show answer and explanation
Answer: B. GCF 2, then x2 + 7x + 12 = (x + 3)(x + 4) because 3 × 4 = 12 and 3 + 4 = 7. A keeps a factor of 2 inside, C stops early, and D gives x2 + 8x + 12.
Factor Completely: frequently asked questions
What does it mean to factor a polynomial completely?
It means to write the polynomial as a product in which no factor can be broken down any more using integer coefficients. You take out the greatest common factor and then factor each remaining piece until every piece is prime. For example, 5x3 − 20x = 5x(x + 2)(x − 2).
How do I know when I am done factoring?
Check each factor. A monomial factor is done. A binomial is done unless it is a difference of squares or has a common factor. A trinomial is done if no pair of integers has the right product and sum. Then multiply everything back to make sure you get the original polynomial.
Does the GCF stay in the answer?
Yes. The greatest common factor is part of the product, so it must appear in the final answer. 3(x + 1)(x − 1) equals 3x2 − 3, but (x + 1)(x − 1) alone equals x2 − 1, a different polynomial. Dropping the GCF is one of the most common lost points on tests.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSA-SSE.A.2
Cite this entry
Effortless Math Team. “Factor Completely.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/factor-completely/
Flashcard
TermFactor Completely2x2 − 18 = ?
DefinitionWrite it as a product where no factor splits further: GCF first, then the rest. 2x2 − 18 = 2(x + 3)(x − 3).
PolynomialA polynomial is an expression built from variables and coefficients using only addition, subtraction, and multiplication, with whole-number exponents. Examples are 3x2 + 5x − 7 and x + 9. Its degree is the highest exponent it contains. A variable in a denominator or under a square root usually means the expression is not a polynomial.Algebra · Algebra 1, Algebra 2CoefficientA coefficient is the number multiplied by a variable in a term. In 4a the coefficient is 4, in −3x it is −3, and in x it is 1, because x = 1x. The sign in front belongs to the coefficient. A number standing alone, with no variable, is a constant instead.Algebra · Grade 6, Grade 7, Grade 8Greatest Common FactorThe greatest common factor (GCF) is the largest factor shared by two or more numbers: the largest number that divides into both without a remainder. The GCF of 18 and 24 is 6. Find it by listing factors or by multiplying the prime factors the numbers share. The GCF is how you simplify fractions in one step.Number System · Grade 5, Grade 6, Grade 7DegreeA degree is the standard unit for measuring an angle, written with the symbol °. One degree is 1360 of a full turn, so a full turn is 360°, a straight angle is 180° and a right angle is 90°. An angle that turns through 25 one-degree angles measures 25°.Geometry · Grade 4, Grade 5, Grade 7NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2SquareA square is a quadrilateral with four equal sides and four right angles. It is a special rectangle and a special rhombus at the same time. For a square with side s, the perimeter is 4s, the area is s2, and each diagonal is s√2.Geometry · Kindergarten, Grade 2, Grade 3Difference of SquaresA difference of squares is an expression of the form a2 − b2: one perfect square subtracted from another. It always factors as (a + b)(a − b). For example, x2 − 49 = (x + 7)(x − 7). A sum of squares, a2 + b2, does not factor over the real numbers.Algebra · Algebra 1, Algebra 2FactoringFactoring rewrites a number or expression as a product of factors. It is the reverse of multiplying or distributing: 12 = 3 × 4, 6x + 9 = 3(2x + 3) and x2 + 5x + 6 = (x + 2)(x + 3). Factoring changes the form, not the value, so multiplying the factors back gives the original.Algebra · Grade 6, Grade 7, Grade 8DifferenceA difference is the result of subtracting one number from another. In 12 − 7 = 5, the difference is 5. It tells how much bigger one number is than another, or how far apart they are on a number line. “The difference of a and b” means a − b, in that order.Operations · Grade 1, Grade 2, Grade 3SimplifySimplify means to rewrite a number or expression in a shorter, easier form without changing its value. To simplify a fraction, divide the numerator and denominator by a common factor until none is left: 1824 = 34. To simplify an expression, combine like terms and do the operations you can: 3x + 2x = 5x.Number System · Grade 4, Grade 5, Grade 6Like TermsLike terms have exactly the same variables raised to exactly the same powers; only their coefficients may differ. 3x and 8x are like terms, and so are 5a2 and −2a2, but 4x and 4y are not. Only like terms can be combined by adding or subtracting their coefficients: 3x + 8x = 11x.Algebra · Grade 6, Grade 7, Grade 8MonomialA monomial is a single term made of a number, a variable, or a product of numbers and variables with whole-number exponents, such as 8, x, or −7x2y. It has no addition or subtraction, and no variable in a denominator. Its degree is the sum of the exponents on its variables.Algebra · Grade 8, Algebra 1BinomialA binomial is a polynomial with exactly two terms, joined by a plus or minus sign. Examples are x + 7, 3m − 4 and a2 − 9. Combine like terms before counting: 3x + 5x is 8x, one term. A polynomial with one term is a monomial; three terms make a trinomial.Algebra · Algebra 1, Algebra 2FactorA factor is a number or expression that divides another number or expression evenly, with no remainder. Factors multiply together to make a product: 4 × 6 = 24, so 4 and 6 are factors of 24. Every whole number greater than 1 has at least two factors, 1 and itself.Number System · Grade 4, Grade 5, Grade 6Prime NumberA prime number is a whole number greater than 1 with exactly two factors: 1 and itself. The first primes are 2, 3, 5, 7, 11 and 13. The number 1 is not prime, and 2 is the only even prime number. Whole numbers greater than 1 that are not prime are composite.Number System · Grade 4, Grade 5, Grade 6Quadratic EquationA quadratic equation is an equation that can be written in the form ax2 + bx + c = 0, where a is not 0. Its graph is a parabola, and it may have zero, one, or two real solutions. You can solve it by factoring, by taking square roots, by completing the square, or with the quadratic formula.Algebra · Algebra 1, Algebra 2TrinomialA trinomial is a polynomial with exactly three terms, such as x2 + 5x + 6 or 2a − 3b + 7. The terms are joined by addition or subtraction and must be unlike terms, so nothing combines. The most common kind is the quadratic trinomial ax2 + bx + c, which students learn to factor into two binomial factors.Algebra · Algebra 1, Algebra 2