बहुपदों का गुणनखंडन

बहुपदों का गुणनखंडन का अर्थ है अभाज्य गुणनखंडन का उपयोग करके दिए गए बहुपद को दो या दो से अधिक बहुपदों के गुणनफल में विघटित करना। बहुपदों का गुणनखंडन बहुपदों को आसानी से सरलीकरण करने में सहायता करता है।

बहुपदों के गुणनखंडन की विधियाँ

पदों को विभाजित करके बहुपदों का गुणनखंडन करना

The process of factoring polynomials is often used for quadratic equations. While factoring polynomials we often reduce the higher degree polynomial into a quadratic expression. Further, the quadratic equation has to be factorized to obtain the factors needed for the higher degree polynomial. The general form of a quadratic equation is $x^{2}+x(a+b)+ab=0$ which can be split into two factors $(x+a)(x+b)=0$

$x^{2}+x(a+b)+ab$

$=x.x+ax+bx+ab$

$=x(x+a)+b(x+a)$

$=(x+a)(x+b)$

In the above polynomial, the middle term is split as the sum of two factors, and the constant term is expressed as the product of these two factors. Thus the given quadratic polynomial is expressed as the product of two expressions.

Example: Let us understand this better, by factoring a quadratic polynomial $x^{2}+7x+12$

$x^{2}+7x+12$

Here the middle term is $7$ and last term is $12$ . The possible combinations of splitting the middle term such that the product of factors of middle term and the last term matching is shown in the below table.

Middle term	Factor 1	Factor 2	Product of factor 1 and factor 2	Last term	Is Product of factor 1 & 2 = Last term
$7$	$1$	$6$	$1\times 6=6$	$12$	No
$7$	$2$	$5$	$2\times 5=10$	$12$	No
$7$	$3$	$4$	$3\times 4=12$	$12$	Yes

Here in the last combination, product of factors of middle term and the last term matches. Hence the factors are $3,4$.

$x.x+3x+4x+3.4$

$x(x+3)+4(x+3)$

$(x+3)(x+4)$

Hence the factors of $x^{2}+7x+12$ = $(x+3)(x+4)$