RD Sharma Chapter 6 Class 9 Maths Exercise 6.4 Solutions is based on the Factorization of Polynomial Factor. According to the factor theorem, let p(x) be a polynomial of degree greater than or equal to one () and a be a real number so that p(a) = 0, then (x – a) is the factor of p(x). In reverse, if (x – a) is a factor of p(x), then p(a) = 0. Important Definitions, Examples, etc., are also mentioned in this article in simple language, which is easy to understand by the students.
Moreover, we have attached the RD Sharma Chapter 6 Class 9 Maths Exercise 6.4 Solutions PDF for the learners from which they may practice for the exam. Professionals prepare the solution given in the PDF with tips, tricks, and shortcut methods. Also, the problems mentioned in the PDF is the combination of the RD Sharma, CBSE Book, and Previous Year Questions of class 9th. We have provided the accessibility to download the PDF so that students can practice offline as well.
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Download RD Sharma Chapter 6 Class 9 Maths Exercise 6.4 Solutions PDF
Solutions for Class 9 Maths Chapter 6 Factorization of Polynomials Exercise 6.4
Important Definition RD Sharma Chapter 6 Class 9 Maths Exercise 6.4 Solutions
As we know, the RD Sharma Chapter 6 Class 9 Maths Exercise 6.4 Solutions is based on the Factorization of Polynomials, in which we will learn about the Factor Theorem.
First, we learn,” What is the Factor Theorem?” So the Factor theorem is generally used for factoring a polynomial and finding the roots of the polynomial. It is also known as the special case of the Remainder Theorem of Polynomial.
According to the Factor Theorem, if p(x) is a polynomial of degree n ≥ 1 and ‘a’ is any real number, then (x-a) is a factor of p(x), if p(a)=0.
Also, we can say, if (x-a) is a factor of polynomial p(x), then p(a) = 0. This explains the converse of the theorem.
How to Prove the Factor Theorem
Considering a polynomial f(x) which is divided by (x-c), then f(c)=0.
Using the remainder theorem,
p(x)= (x-c) q(x)+p(c)
Where p(x) is the target polynomial and q(x) is the quotient polynomial.
Since, p(c) = 0, hence,
p(x)= (x-c) q(x)+ p(c)
p(x) = (x-c) q(x)+ 0
p(x) = (x-c) q(x)
Therefore, (x-c) is the factor of the polynomial p(x).
We can prove the factor theorem using the Remainder Theorem, which is the Alternate Method of proving the Polynomial factor. Check below the Alternate Method by using the Remainder Method-
p(x)= (x-c)q (x)+ p(c)
If (x-c) is a factor of p(x), then the remainder should be zero.
(x-c) exactly divides p(x)
Therefore, p(c)=0.
The following remarks are equivalent for any polynomial p(x)-
- The remainder is zero when p(x) is exactly divided by (x-c).
- (x-c) is a factor of p(x).
- c is the solution to p(x).
- c is a zero of the function p(x), or p(c) =0.
Steps to Use Factor Theorem
Know how to find the factors of the polynomial by using Factor Theorem-
Step 1- If f(-c) = 0, then (x+ c) is a factor of the polynomial f(x).
Step 2- If p(d/c) = 0, then (cx-d) is a factor of the polynomial f(x).
Step 3- If p(-d/c) = 0, then (cx+d) is a factor of the polynomial f(x).
Step 4- If p(c) = 0 and p(d) = 0, then (x-c) and (x-d) are factors of the polynomial p(x).
Frequently Asked Questions (FAQs) of RD Sharma Chapter 6 Class 9 Maths Exercise 6.4 Solutions
Ques 1- What is the difference between the Factor Theorem and the Remainder Theorem?
Ans- The remainder theorem shows that for any polynomial p(x), if we divide it by the binomial x−a, the remainder is equivalent to the value of p(a).
Whereas, the factor theorem shows that if ‘a’ is a zero of a polynomial p(x), so the (x−a) is a factor of p(x), and vice-versa.
Ques 2- What are the factors of 2x?
Ans- 2 and x are the factors of 2x.