Factorization

of Polynomials

What is Factorization?

factorization or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 × 5, and the polynomials x2 − 4 factors as (x − 2)(x + 2). In all cases, a product of simpler objects is obtained.

Prime Factorization

Integer factorization or Prime factorization is the breaking down of a composite number into smaller non-trivial divisors, which when multiplied together equal the original integer.

Perfect square trinomials

Some quadratics can be factored into two identical binomials. These quadratics are called perfect square trinomials. Perfect square trinomials can be factored as follows:

a2+2ab+b2=(a+b) 2 a2-2ab+b2=(a-b) 2

Sum/difference of two squares

Another common type of algebraic factoring is called the difference of two squares. It is the application of the formula:

a2-b2=(a+b)(a-b)

to any two terms, whether or not they are perfect squares. If the two terms are subtracted, simply apply the formula. If they are added, the two binomials obtained from the factoring will each have an imaginary term. This formula can be represented as

a2+b2=(a+bi)(a-bi)

Factoring by grouping

Another way to factor some polynomials is factoring by grouping. This is done by placing the terms in the polynomial into two or more groups, where each group can be factored by a known method. The results of these factorizations can sometimes be combined to make an even more simplified expression. For example, to factor the polynomial

4x2+20x + 3yx+15y

Group similar terms: (4x2 +20x) + (3yx+15y)

Factor out greatest common factor: 4x(x+5)+3y(x+5)

Factor out binomial: (x+5)(4x+3y)

Factoring other polynomials

Sum/difference of two cubes

Another formula for factoring is the sum or difference of two cubes. The sum can be represented by (a3 + b3)= (a + b)(a2 –ab + b2)

and the difference by(a3 - b3)=(a - b)(a2 –ab+ b2)

For example, x3 − 103 (or x3 − 1000) can be factored into (x − 10)(x2 + 10x + 100).

Definition of terms Prime number is a natural number that has exactly

two distinct natural number divisors 1 and itself . Polynomial is an expression of finite length

constructed from variables (also known as indeterminate) and constants, using only the operations of addition, subtraction, multiplication, and non-negative, whole-number exponents. For example, x2 − 4x + 7 is a polynomial, but x2 − 4/x + 7x3/2 is not, because its second term involves division by the variable x and because its third term contains an exponent that is not a whole number.

Definition of terms

Composite number is a positive integer which has a positive divisor other than one or itself.

Binomial is a polynomial with two terms—the sum of two monomials—often bound by parenthesis or brackets when operated upon. It is the simplest kind of polynomial.

Quadratic describes something that pertains to squares, to the operation of squaring, to terms of the second degree, or equations or formulas that involve such terms. Quadratus is Latin for square.