बहुपदों का गुणनखंडन: Difference between revisions

From Vidyalayawiki

(added content)
Line 10: Line 10:


=== पदों को विभाजित करके बहुपदों का गुणनखंडन करना ===
=== पदों को विभाजित करके बहुपदों का गुणनखंडन करना ===
=== Factoring Polynomials by Splitting Terms ===
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 <math>x^2+x(a+b)+ab=0</math> which can be split into two factors <math>(x+a)(x+b)=0</math>
<math>x^2+x(a+b)+ab</math>
<math>=x.x+ax+bx+ab</math>
<math>=x(x+a)+b(x+a)</math>
<math>=(x+a)(x+b)</math>
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 <math>x^2+7x+12</math>
<math>x^2+7x+12</math>
Here the middle term is  <math>7</math> and last term is <math>12</math> . 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.
{| class="wikitable"
|+
|'''Middle term'''
|'''Factor 1'''
|'''Factor 2'''
|'''Product of factor 1 and factor 2'''
|'''Last term'''
|'''Is Product of factor 1 & 2 = Last term'''
|-
|<math>7</math>
|<math>1</math>
|<math>6</math>
|<math>1 \times 6 = 6</math>
|<math>12</math>
|No
|-
|<math>7</math>
|<math>2</math>
|<math>5</math>
|<math>2 \times 5 = 10</math>
|<math>12</math>
|No
|-
|<math>7</math>
|<math>3</math>
|<math>4</math>
|<math>3 \times 4 = 12</math>
|<math>12</math>
|Yes
|}
Here in the last combination, product of factors of middle term and the last term matches. Hence the factors are <math>3,4</math>.
<math>x.x+3x+4x+3.4</math>
<math>x(x+3)+4(x+3)</math>
<math>(x+3)(x+4)</math>
Hence the factors of <math>x^2+7x+12</math> = <math>(x+3)(x+4)</math>

Revision as of 16:37, 15 May 2024

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 का गुणनखंडन क्या है?

बहुपदों के गुणनखंडन की प्रक्रिया

बहुपदों के गुणनखंडन की विधियाँ

पदों को विभाजित करके बहुपदों का गुणनखंडन करना

Factoring Polynomials by Splitting Terms

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 which can be split into two factors

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

Here the middle term is and last term is . 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
No
No
Yes

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

Hence the factors of =