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what is common monomial factor

A common monomial factor is a single-term expression (a monomial) that divides every term of a given algebraic expression without leaving a remainder.

For example, in the expression 24x6−12x4−18x324x^6-12x^4-18x^324x6−12x4−18x3, the terms are 24x624x^624x6, 12x412x^412x4, and 18x318x^318x3.

Each term is divisible by 6x36x^36x3, so 6x36x^36x3 is a common monomial factor (in fact, it’s the greatest one).

Quick Scoop: Key Idea

  • A monomial is a single term like 5x5x5x, −3y2-3y^2−3y2, or 7a3b7a^3b7a3b.
  • A common factor is something that divides all terms in an expression.
  • Put together, a common monomial factor is a monomial that is a factor of each term in a polynomial.

You usually look for:

  • A number that divides all coefficients (like 2, 3, 6, etc.).
  • Variables with powers that appear in every term (taking the lowest exponent of each common variable).

How to Find the Common Monomial Factor

You can think of it as a 3-step routine:

  1. Break each term into factors
    • Example: 24x6=2⋅2⋅2⋅3⋅x⋅x⋅x⋅x⋅x⋅x24x^6=2\cdot 2\cdot 2\cdot 3\cdot x\cdot x\cdot x\cdot x\cdot x\cdot x24x6=2⋅2⋅2⋅3⋅x⋅x⋅x⋅x⋅x⋅x.
 * Do this for all terms.
  1. Pick what’s common to all terms
    • Choose the largest number that divides all coefficients (greatest common factor of the numbers).
 * For variables, choose each variable with the _smallest_ exponent that appears in every term.
  1. Multiply those common parts together
    • That product is your common monomial factor (often you choose the greatest one so factoring is most effective).

Example Walkthrough

Take the polynomial:

24x6−12x4−18x324x^6-12x^4-18x^324x6−12x4−18x3

  1. Coefficients: 24, 12, 18
    • Greatest common factor of 24, 12, 18 is 6.
  1. Variables: x6,x4,x3x^6,x^4,x^3x6,x4,x3
    • Common variable is xxx, and the smallest exponent among 6, 4, 3 is 3, so variable part is x3x^3x3.
  1. Combine: common monomial factor is 6x36x^36x3.

If you factor it out:

24x6−12x4−18x3=6x3(4x3−2x−3)24x^6-12x^4-18x^3=6x^3(4x^3-2x-3)24x6−12x4−18x3=6x3(4x3−2x−3)

Here, 6x36x^36x3 is the common monomial factor you pulled out.

Why It Matters (In Today’s Math Class 😄)

  • It’s the first step in most factoring problems in algebra, especially when working with polynomials.
  • It simplifies expressions and prepares them for more advanced factoring like trinomials, special products, and solving equations.

In one line: A common monomial factor is a single-term expression (like 6x36x^36x3) that divides every term in a polynomial, and we usually choose the largest such monomial to factor expressions efficiently.

TL;DR:
If all terms share the same “number × variables,” that shared piece is the common monomial factor (for example, 2xy2xy2xy in −8xy2+12xy+18x2y-8xy^2+12xy+18x^2y−8xy2+12xy+18x2y has 2xy2xy2xy in every term).

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