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Factor of a Polynomial

Factor of a Polynomial
Factorization of a Polynomial

A factor of polynomial P(x) is any polynomial which divides evenly into P(x). For example, x + 2 is a factor of the polynomial x2 – 4.

The factorization of a polynomial is its representation as a product its factors. For example, the factorization of x2 – 4 is (x – 2)(x + 2).

 

 

See also

Polynomial long division, synthetic division

Key Formula

P(x)=F1(x)⋅F2(x)⋯Fn(x)P(x) = F_1(x) \cdot F_2(x) \cdots F_n(x)
Where:
  • P(x)P(x) = The original polynomial being factored
  • F1(x),F2(x),…,Fn(x)F_1(x), F_2(x), \ldots, F_n(x) = The polynomial factors whose product equals P(x)

Worked Example

Problem: Find all factors of the polynomial P(x) = x³ − 7x + 6.
Step 1: Test possible rational roots using the Factor Theorem: if P(c) = 0, then (x − c) is a factor. Try x = 1.
P(1)=1−7+6=0P(1) = 1 - 7 + 6 = 0
Step 2: Since P(1) = 0, we know (x − 1) is a factor. Divide P(x) by (x − 1) using synthetic or long division.
x3−7x+6=(x−1)(x2+x−6)x^3 - 7x + 6 = (x - 1)(x^2 + x - 6)
Step 3: Factor the remaining quadratic x² + x − 6. Find two numbers that multiply to −6 and add to 1: those are 3 and −2.
x2+x−6=(x+3)(x−2)x^2 + x - 6 = (x + 3)(x - 2)
Step 4: Write the complete factorization by combining all factors.
x3−7x+6=(x−1)(x+3)(x−2)x^3 - 7x + 6 = (x - 1)(x + 3)(x - 2)
Answer: The factors of x³ − 7x + 6 are (x − 1), (x + 3), and (x − 2).

Another Example

Problem: Show that (x − 4) is a factor of P(x) = x² − x − 12.
Step 1: Use the Factor Theorem: evaluate P(4). If the result is 0, then (x − 4) is a factor.
P(4)=16−4−12=0P(4) = 16 - 4 - 12 = 0
Step 2: Since P(4) = 0, confirm (x − 4) is a factor and find the other factor by factoring the quadratic.
x2−x−12=(x−4)(x+3)x^2 - x - 12 = (x - 4)(x + 3)
Answer: Because P(4) = 0, (x − 4) is indeed a factor. The full factorization is (x − 4)(x + 3).

Frequently Asked Questions

How do you find the factors of a polynomial?
For quadratics, look for two numbers that multiply to give the constant term (times the leading coefficient, if it isn't 1) and add to the middle coefficient. For higher-degree polynomials, use the Factor Theorem: test values c where P(c) = 0, which tells you (x − c) is a factor, then divide the polynomial by that factor and repeat on the quotient.
What is the difference between a factor and a root of a polynomial?
A root (or zero) is a number c such that P(c) = 0. A factor is the corresponding binomial (x − c). By the Factor Theorem, c is a root of P(x) if and only if (x − c) is a factor of P(x). So roots are numbers; factors are expressions.

Factor of a Polynomial vs. Root (Zero) of a Polynomial

A factor is a polynomial expression like (x−3)(x - 3) that divides evenly into P(x)P(x). A root is the value x=3x = 3 that makes P(x)=0P(x) = 0. They are two sides of the same coin: the Factor Theorem states that (x−c)(x - c) is a factor of P(x)P(x) exactly when cc is a root of P(x)P(x).

Why It Matters

Factoring polynomials is one of the most powerful tools in algebra. It lets you solve polynomial equations by setting each factor equal to zero, which is far simpler than working with the original expression. Factoring also appears in simplifying rational expressions, graphing polynomial functions, and throughout calculus when finding critical points.

Common Mistakes

Mistake: Forgetting that a constant like 3 can be a factor of a polynomial.
Correction: Constants count as factors too. For example, 3x² + 6x = 3(x² + 2x) = 3x(x + 2). Always check for a greatest common factor (GCF) before attempting other methods.
Mistake: Confusing 'factor' with 'term' — treating each separate piece like x² and −4 in x² − 4 as factors.
Correction: Terms are the parts of a polynomial separated by addition or subtraction. Factors are expressions that multiply together to produce the polynomial. In x² − 4, the terms are x² and −4, but the factors are (x − 2) and (x + 2).

Related Terms