Taking Out a Common Factor

1 MIN READ · PRACTISE AT YOUR OWN PACE · FREE

Idea

A common factor appears in every term. For integer coefficients, take their greatest common divisor; for each variable, take the smallest exponent appearing in all terms. Distributing the result is the quickest way to check it.

Visual guide

VISUAL GUIDEOne height, two widths

The shared height 3 makes the total area 3x + 6 equal to 3(x + 2). The picture uses x = 4; the factorization holds for all real x.

Method

The distributive law works in reverse:

ab+ac=a(b+c).

A polynomial factor can also be common. Keep its parentheses intact while treating it as one object.

Worked example

In $18x^3y-12x^2y^2$ , the coefficient factor is $6$ , and the shared variable part is $x^2y$ . Thus

18x^3y-12x^2y^2=6x^2y(3x-2y).

For $x(x+1)+4(x+1)$ , the repeated object is $x+1$ , giving $(x+1)(x+4)$ .

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

Factor

6x^2+9x

completely.

$3x(2x+3)$

$3(2x+3)$

$3x(2x+9)$

Hint 1 · Find a starting point

Find the greatest factor common to both terms.

Hint 2 · Take the next step

Divide both terms by 3x, retaining the factor outside.

Show the reasoning

Answer: $3x(2x+3)$

Expanding 3x(2x+3) returns 6x²+9x.

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

Common mistake

Do not take a variable outside if it is absent from one term. For example, $x^2+3$ has no common factor $x$ as a polynomial.

Check your understanding

Factor $15a^2b+10ab^2$ .

Show answer

$5ab(3a+2b)$ . Multiplication gives the original terms.

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

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