RD SHARMA SOLUTION CHAPTER- 6 Factorization of Polynomials | CLASS 9TH MATHEMATICS-EDUGROWN

Chapter 6 Factorisation of Polynomials Exercise Ex. 6.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8Give one example each of a binomial of degree 35, and of a monomial of degree 100.

Solution 8Degree of a polynomial is the highest power of variable in the polynomial.
Binomial has two terms in it. So binomial of degree 35 can be written as x³⁵ + 7 .  Monomial has only one term in it. So monomial of degree 100 can be written as 7x¹⁰⁰. Concept Insight: Mono, bi and tri means one, two and three respectively. So, monomial is a polynomial having one term similarly for binomials and trinomials. Degree is the highest exponent of variable.  The answer is not unique in such problems . Remember that the terms are always separated by +ve or -ve sign and not with  .  

Chapter 6 Factorisation of Polynomials Exercise Ex. 6.2

Question 1If f(x) = 2x³ – 13x² + 17x + 12, find:

(i) f(2)

(ii) f(-3)

(iii) f(0)Solution 1(i)

f(x) = 2x³ – 13x² + 17x + 12

f(2) = 2(2)³ – 13(2)² + 17(2) + 12

      = 16 – 52 + 34 + 12

      = 10

(ii)

f(-3) = 2(-3)³ – 13(-3)² + 17(-3) + 12

       = -54 – 117 – 51 + 12

       = – 210

(iii)

f(0) = 2(0)³ – 13(0)² + 17(0) + 12

      = 0 – 0 + 0 + 12

      =12Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7Find rational roots of the polynomial f(x) = 2x³ + x² – 7x – 6.Solution 7

Chapter 6 Factorisation of Polynomials Exercise Ex. 6.3

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

If the polynomials ax³ + 3x² – 13 and 2x³ -5x + a, when divided by (x-2) leave the same remainder, find the value of a.Solution 10

Question 11Find the remainder when x³ + 3x² + 3x + 1 is divided by 

Solution 11

Question 12

Solution 12

Chapter 6 Factorisation of Polynomials Exercise Ex. 6.4

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

What must be added to 3x³ + x² – 22x + 9 so that the result is exactly divisible by 3x² + 7x – 6?Solution 25

Chapter 6 Factorisation of Polynomials Exercise Ex. 6.5

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Using factor theorem, factorize: x⁴ – 7x³ + 9x² + 7x -10Solution 4

Question 5

Using factor theorem, factorize: 3x³ – x² – 3x + 1Solution 5

Question 6

Using factor theorem, factorize each of the following polynomials:

x³ – 23x² + 142x – 120Solution 6

Question 7

Using factor theorem, factorize: y³ – 7y + 6Solution 7

Question 8

Using factor theorem, factorize: x³ – 10x² – 53x – 42Solution 8

Question 9

Using factor theorem, factorize: y³ – 2y² – 29y – 42Solution 9

Question 10

Using factor theorem, factorize: 2y³ – 5y² – 19y + 42Solution 10

Question 11

x³ + 13x² + 32x + 20Solution 11

         Let p(x) = x³ + 13x² + 32x + 20
         The factors of 20 are 1,  2,  4,  5 … …
         By hit and trial method
         p(- 1) = (- 1)³ + 13(- 1)² + 32(- 1) + 20
                   = – 1 + 13 – 32 + 20
                   = 33 – 33 = 0
         As p(-1) is zero, so x + 1 is a factor of this polynomial p(x).

         Let us find the quotient while dividing x³ + 13x² + 32x + 20 by (x + 1)
          By long division

                   We know that
         Dividend = Divisor  Quotient + Remainder
         x³ + 13x² + 32x + 20 = (x + 1) (x² + 12x + 20) + 0
                                         = (x + 1) (x² + 10x + 2x + 20)
                                         = (x + 1) [x (x + 10) + 2 (x + 10)]
                                         = (x + 1) (x + 10) (x + 2)
                                         = (x + 1) (x + 2) (x + 10)Question 12

Factorise:

x³ – 3x² – 9x – 5Solution 12

        Let p(x) = x³ – 3x² – 9x – 5
        Factors of 5 are 1,  5.
        By hit and trial method
        p(- 1) = (- 1)³ – 3(- 1)² – 9(- 1) – 5
           = – 1 – 3 + 9 – 5 = 0
        So x + 1 is a factor of this polynomial
        Let us find the quotient while dividing x³ + 3x² – 9x – 5 by x + 1
        By long division

                Now, Dividend = Divisor  Quotient + Remainder
 x³ – 3x² – 9x – 5 = (x + 1) (x² – 4 x – 5) + 0
                                     = (x + 1) (x² – 5 x + x – 5)
                                     = (x + 1) [(x (x – 5) +1 (x – 5)]
                                     = (x + 1) (x – 5) (x + 1)
                                     = (x – 5) (x + 1) (x + 1) Question 13

Factorise:

2y³ + y² – 2y – 1Solution 13

         Let p(y) = 2y³ + y² – 2y – 1         By hit and trial method
         p(1) = 2 ( 1)³ + (1)² – 2( 1) – 1
                = 2 + 1 – 2 – 1= 0
         So, y – 1 is a factor of this polynomial
         By long division method,

                    p(y) = 2y³ + y² – 2y – 1
                 = (y – 1) (2y² +3y + 1)
                 = (y – 1) (2y² +2y + y +1)
                 = (y – 1) [2y (y + 1) + 1 (y + 1)]
                 = (y – 1) (y + 1) (2y + 1)

 Question 14

Using factor theorem, factorize: x³ – 2x² – x + 2Solution 14

Question 15

Solution 15

Question 16

Using factor theorem, factroize : x⁴ – 2x³ – 7x² + 8x + 12Solution 16

Question 17

Using factor theorem, factroize : x⁴ + 10x³ + 35x² + 50x + 24Solution 17

Question 18

Using factor theorem, factorize : 2x⁴ – 7x³ – 13x² + 63x – 45Solution 18