### Topic Page: factor

**factor**from

*The Columbia Encyclopedia*

in arithmetic, any number that divides a given number evenly, i.e., without any remainder. The factors of 12 are 1, 2, 3, 4, 6, and 12. Similarly in algebra, any one of the algebraic expressions multiplied by another to form a product is a factor of that product, e.g., a+b and a-b are factors of a^{2}-b^{2}, since (a+b)(a-b)=a^{2}-b^{2}. In general, if r is a root of a polynomial equation f(x)=0, then (x-r) is a factor of the polynomial f(x).

**factor**from

*The Hutchinson Unabridged Encyclopedia with Atlas and Weather Guide*

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 2a^{2} + 6ab are 2a and a + 3b, since 2a^{2} + 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 24x^{2} − 16x. 24 and 16 have a highest common factor of 8 so

24x^{2} − 16x = 8(3x^{2} − 2x)

3x^{2} and 2x have a common factor of x so

24x^{2} − 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 x^{2} + 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 x^{2} + 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 +:

x^{2} + 9x + 20 = (x + ?)(x + ?)

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

x^{2} + 9x + 20 = (x + 5)(x + 4)

3. To factorize x^{2} − 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 +:

x^{2} − 7x + 10 = (x − ?)(x − ?)

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

x^{2} − 7x + 10 = (x − 2)(x − 5)

4. To factorize x^{2} + 10x − 24. The number at the end is −24, so the numbers must have different signs so that they multiply to give a −:

x^{2} + 10x − 24 = (x + ?)(x − ?)

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

x^{2} + 10x − 24 = (x + 12)(x − 2)

The above methods apply if the quadratic is of the form x^{2} + bx + c. If the quadratic is of the form ax^{2} + bx + c, factorization is more complex and must take account of the value of the constant a; for example:

3x^{2} + 7x + 2 = (3x + 1)( x + 2)

essays

One from the Top Please Carol – Using the Four Rules of Number

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