factor

Number that divides into another number exactly. It is also known as a divisor. For example, the factors of 24 are 1, 2, 4, 8, 12, and 24; and the factors of 64 are 1, 2, 4, 8, 16, 32, and 64. The highest factor of both 24 and 64 is 8. This is known as the highest common factor (HCF) of the two numbers.

The largest number to be factorized to date has 167 digits: it was split into its 80- and 87-digit factors in 1997 by US mathematicians, after 100,000 hours of computing.

See also prime number and prime factor.

factor

In algebra, certain kinds of polynomials (expressions consisting of several or many terms) can be factorized using their common factors. Brackets are put into an expression, and the common factor is sought. For example, the factors of 2a² + 6ab are 2a and a + 3b, since 2a² + 6ab = 2a(a + 3b). This rearrangement is called factorization.

Factorization The first thing to look for is a common factor. For example:

1. To factorize 6a + 9b. The numbers 6 and 9 both have a common factor of 3. 3 is the biggest number that divides exactly into 6 and 9. The 3 is taken outside a bracket as a factor:

6a + 9b = 3(   )

and the content of the bracket is worked out:

3 × 2a = 6a and 3 × 3b = 9b

This means that 6a + 9b = 3(2a + 3b) when factorized.

2. To factorize 24x2 − 16x. 24 and 16 have a highest common factor of 8 so

24x2 − 16x = 8(3x2 − 2x)

3x2 and 2x have a common factor of x so

24x2 − 16x = 8x(3x − 2)

In this example the factorization could be done all at once, as the common factor is 8x.

Factorization of quadratic expressions A quadratic expression can be factorized by splitting the expression into two brackets. For example:

1. To factorize x2 + 3x + 2, split the expression into two brackets:

(x + 2)(x + 1)

the 2 is found by multiplying the two numbers at the end of the brackets together:

2 × 1 = 2

The coefficient of x, in this case 3, is found by adding the two numbers at the end of the bracket together:

2 + 1 = 3

It is often helpful to look at the signs in the quadratic expression to be factorized.

2. To factorize x2 + 9x + 20. The number at the end is +20, so the numbers in the brackets must be the same sign so that they can multiply to give a +:

x2 + 9x + 20 = (x + ?)(x + ?)

The signs must both be + because the numbers must add to give +9:

x2 + 9x + 20 = (x + 5)(x + 4)

3. To factorize x2 − 7x + 10. The number at the end is +10, so the numbers in the brackets must be the same sign so that they multiply to give a +:

x2 − 7x + 10 = (x − ?)(x − ?)

They must both be − because they add to give −7:

x2 − 7x + 10 = (x − 2)(x − 5)

4. To factorize x2 + 10x − 24. The number at the end is −24, so the numbers must have different signs so that they multiply to give a −:

x2 + 10x − 24 = (x + ?)(x − ?)

The + number must be bigger because they must add to give a total of +10:

x2 + 10x − 24 = (x + 12)(x − 2)

The above methods apply if the quadratic is of the form x2 + bx + c. If the quadratic is of the form ax2 + bx + c, factorization is more complex and must take account of the value of the constant a; for example:

3x2 + 7x + 2 = (3x + 1)( x + 2)