RD Sharma Class 9 Solutions Chapter 6 - Factorization of Polynomials (Ex 6.4) Exercise 6.4 - Free PDF
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Factorization of polynomials is considered the most important topic for class 9 students in all of their exams. In further topics, one of the primary steps would be the factorization of a polynomial. If you can't factorise a polynomial, you won’t be able to solve any questions further. So, it is advised to students to practice this topic well before moving any further.
As we already know, factorization is a process of finding the factors of a given mathematical value (be it a number, a polynomial or an algebraic expression) and these factors, when multiplied, give the same output as the mathematical value itself. Factorization of polynomials is similar to factorization of any numerical values. We determine all of the terms that were multiplied together to get the given polynomial as a result. We then try to factorise each of the terms we found in the first step. This continues until we simply can’t factorize the polynomial any further. When we can’t do any more factoring we will say that the polynomial is completely done with factorization.
Process of Factorization of Polynomials
The following steps can help in the process of factorization of polynomials. Follow the below steps to factorize a polynomial easily.
Factor out if there is a common factor to all the terms of the given polynomial.
Identify a correct and appropriate method for the factorization of polynomials. Students can use two methods that are by regrouping and algebraic identities for finding the factors.
Write the polynomials as the product of their factors.
Methods of Factoring Polynomials
There are numerous methods for factorization of polynomials, based on the given expression. The method used for factorization will depend on the degree of the polynomials and the number of variables present in it. The four important and efficient methods of factorization of polynomials are as follows:
Factorization of polynomials using Method of Common Factors
Factorization of polynomials using Grouping Method
Factorization of polynomials by splitting terms
Factorization of polynomials using Algebraic Identities
Factorization of Polynomials using Method of Common Factors
Method of common factors is the simplest method of factorization of a given polynomial. It is done by choosing the common factors of each term of the given polynomial. Firstly, the factors of each of the terms of the given polynomial are written. Then, the common factors across the given terms of the polynomial are taken to obtain the possible factors. This can also be called an equivalent to using the distributive property in reverse.
Factorization of Polynomials using Grouping Method
The method of grouping for factorization of polynomials is like a continuation step to the method of finding common factors. Here, we try to find groups from the common factors, to get the factors of the given polynomial expression as a result. The number of terms of the given polynomial expression is reduced to a comparatively lesser number of groups. Firstly, we split each of the terms of the given polynomial expression into different factors and by choosing the common terms to form a group of required factors.
Factorization of Polynomials by Splitting Terms
The process of factorization of polynomials is often used for quadratic equations. While factorizing the given polynomials we often reduce the higher degree polynomial into a quadratic expression. Then, the quadratic equation has to be factorized to obtain the factors needed for the higher degrees of polynomials.
Factoring Polynomials using Algebraic Identities
The process of factorization of polynomials can be done by using algebraic identities. The available polynomial expressions represent one of the algebraic identities that we know. Sometimes, the available expression has to be changed so that it matches with the expression of the algebraic identities.
FAQs on RD Sharma Class 9 Solutions Chapter 6 - Factorization of Polynomials (Ex 6.4) Exercise 6.4
1. What is the Remainder theorem?
The remainder theorem is helpful in finding the remainder on dividing an algebraic expression with another expression, without actually performing the division. The remainder that we obtain when the algebraic expression f(x) is divided by (x - n) is f(n). If f(n) = 0, then (x - n) is a factor of f(x).
2. What is the Factor theorem?
The factor theorem efficiently helps in connecting the factors and zeros of the given polynomials. If there is a polynomial of degree n, m is a real number such that (x - m) is a factor of f(x), then if f(m) = 0. Also, f(m) = 0 then (x - m) is a factor of f(x). The factor theorem is very useful for finding the given expression as a factor of a higher degree polynomial expression without doing the division.
3. What is the Long Division?
The process called the long division involving polynomials is very similar to the process of the long division of natural numbers. Long division of polynomials is greatly helpful and efficient to find the factors of the given polynomial expression. The division which is resulting in a remainder of zero has the divisor as a factor of the given polynomial expression. Division occurring in the remainder of zero is written as follows:
Dividend = Divisor × Quotient.
Thus, the given polynomial expression gets divided into two main factors.
4. How to obtain Greatest common factors?
The process of obtaining the greatest common factor for two or more terms of the given polynomial includes two simple steps. Firstly, we split each of the terms into its prime factors. Then, take as many common factors that are possible from the given terms.
5. How to factorise polynomials in 3 degrees?
The process of factorization of polynomials in 3 degrees involves three very simple steps. Firstly, for the given n degree polynomial f(x), substitute a value 'm' such that f(m) = 0, and (x - m) is a factor. In the second part, divide f(x) by (x - m) to obtain a quadratic equation. Finally, we do factorization of the quadratic equation to obtain its two factors. Hence, we can obtain all three factors of the 3-degree polynomial. Visit the official website of Vedantu or log on to the app for a detailed explanation of all of the above.