NCERT Solutions for Class 9 Maths Chapter 2 - Polynomials

Chapter 2 - Polynomials Exercise Ex. 2.1

Solution 1
(i)     Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials

         Yes, this expression is a polynomial in one variable x.

(ii)     Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials

          Yes, this expression is a polynomial in one variable y.

(iii)     Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials

           No, here the exponent of variable t in term Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials which is not a whole number.

           So this expression is not a polynomial.

(iv)     Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials

          No, here the exponent of variable t in term Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials is -1, which is not a whole number. So this expression is not a polynomial.

(v)     Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials

          No, this expression is a polynomial in 3 variables x, y and t.

Concept Insight: In such problems to check whether the given algebraic expressions is a polynomial or not, check the exponents of variable to be a whole number. The second step is to look for the number of variables the expression has. Any alphabet used in the expression is the variable unless specified as constant.

 

Solution 2
(i)     Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
         In the above expression coefficient of x2 is 1
 
(ii)     Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
          In the above expression coefficient of x2 is - 1.
 
(iii)     Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
           In the above expression coefficient of  Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
(iv)     Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
 
            In the above expression coefficient of x2  is 0.
 
Concept Insight: The constant/ variable multiplied with the variable is the coefficient of the variable. Also consider the sign (+ve or the -ve) of the term while writing the coefficient. 
Solution 3
Degree 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  .  
Solution 4
 Degree of a polynomial is the highest power of variable in the polynomial.
 
Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
          This is a polynomial in variable x and highest power of variable x is 3. So, degree of
          this polynomial is 3
 
Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
           This is a polynomial in variable y and the highest power of variable y is 2. So degree
           of this polynomial is 2.
Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
           This is a polynomial in variable t and the highest power of variable t is 1. So degree
           of this polynomial is 1.
(iv)      3
           This is a constant polynomial. Degree of a constant polynomial is always 0.
 
Concept Insight: Degree is the highest  exponent of the variable. While finding the degree of a polynomial express the polynomial in standard form  Combine the like terms and Remember the result that xo = 1.
Solution 5
(i)       2 + x2 + x  is a quadratic polynomial as its degree is 2.

(ii)      x - x3 is a cubic polynomial as  its degree is 3.

(iii)     y + y2 + 4  is a quadratic polynomial as its degree is 2.

(iv)     1 + x is a linear polynomial as its degree is 1.

(v)      3t is a linear polynomial as its degree is 1.

(vi)     r2 is a quadratic polynomial as its degree is 2.

(vii)     7x3 is a cubic polynomial as its degree is 3.
 
 
Concept Insight: Linear polynomial, quadratic polynomial and cubic polynomial has its degrees as 1, 2, and 3 respectively.

Chapter 2 - Polynomials Exercise Ex. 2.2

Solution 1
Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
Concept Insight: Given Polynomial is p(x) to find the  value of given polynomial at any particular value of x replace the variable x with its corresponding value. Remember for odd power of negative number the negative sign remains while for even power of negative numbers the negative sign vanishes.  Also check for  calculation errors; there are chances of making calculation mistake while computing square, cubes and higher powers of numbers.  
Solution 2
(i)       p(y) = y2 - y + 1
          p(0) = (0)2 - (0) + 1 = 1
          p(1) = (1)2 - (1) + 1 = 1
          p(2) = (2)2 - (2) + 1 = 3
 
(ii)      p(t) = 2 + t + 2t2 - t3
          p(0) = 2 + 0 + 2 (0)2 - (0)3  = 2
          p(1) = 2 + (1) + 2(1)2 - (1)3
                 = 2 + 1 + 2 - 1 = 4
          p(2) = 2 + 2 + 2(2)2 - (2)3
                 = 2 + 2 + 8 - 8 = 4
 
(iii)     p(x) = x3     
          p(0) = (0)3 = 0
          p(1) = (1)3 = 1
          p(2) = (2)3 = 8
 
(iv)     p(x) = (x - 1) (x + 1)
          p(0) = (0 - 1) (0 + 1) = (- 1) (1) = - 1
          p(1) = (1 - 1) (1 + 1) = 0 (2) = 0
          p(2) = (2 - 1 ) (2 + 1) = 1(3) = 3
 
Concept Insight: Replace the variable with 0, 1 or  2 in the given polynomials to obtain the required value.  Be careful about the calculations, there are chances of making calculation mistake while computing square, cubes and higher powers of numbers. Carefully apply the properties of addition, subtraction and multiplication of numbers. While multiplying two binomials multiply each term of the binomial to each term of the other binomial.
Solution 3
(i)    If Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials is a zero of given polynomial p(x) = 3x + 1, then
 
        Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
        So, Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials is a zero of given polynomial
 
(ii)    If Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials is a zero of polynomial p(x) = 5x - Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials ,                                          
        then Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials should be 0
 
        Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
         So, Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials is not a zero of given polynomial
 
(iii)    If x = 1 and x = - 1 are zeroes of polynomial p(x) = x2 - 1 then p(1) and p(- 1) 
         should be 0
         Now, p(1) = (1)2 - 1 = 0
         p(- 1) = (- 1)2 - 1 = 0
         Hence x = 1 and - 1 are zeroes of polynomial.
(iv)    If x = - 1 and x = 2 are zeroes of polynomial p(x) = (x +1) (x - 2), then p(- 1) and
         p(2)should be 0.
         Now, p(- 1) = (- 1 + 1) (- 1 - 2) = 0 (-3) = 0
         p(2) = (2 + 1) (2 -  2 ) = 3 (0) = 0
         So, x = - 1 and x = 2 are zeroes of given polynomial.
 
(v)     If x = 0 is a zero of polynomial p(x) = x2 then p(0) should be zero.
         Now, p(0) = (0)2 = 0
         Hence x = 0 is a zero of given polynomial
 
(vi)    If Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials is a zero of polynomial p(x) = lx + m, thenNcert Solutions Cbse Class 9 Mathematics Chapter - Polynomialsis 0.
 
         Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
(vii)    If Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials and Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials are zeroes of polynomial p(x) = 3x2 - 1, then
 
 
          Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
 
            Hence, Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials is a zero of given polynomial but Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials is not a zero of given
 
 
            polynomial.
 
(viii)      If  Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials  is a zero of polynomial p(x) = 2x + 1 then Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials should be 0.
 
 
            Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
            So, Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials is not a zero of polynomial.
 
 
Concept Insight: Key idea here is  Zero of the polynomial is not the real number zero but it is that value of the variable which makes the  value of the polynomial equal to zero. A polynomial can have more than one zeroes.
Solution 4
Zero of a polynomial is that value of variable at which value of polynomial comes to 0.
(i)     p(x) = x + 5
        p(x) = 0
        x + 5 = 0
        x = - 5
        So, for x = - 5, value of polynomial is 0 and hence x = - 5 is a zero of polynomial.
(ii)    p(x) = x - 5
        p(x) = 0
        x - 5 = 0
        x = 5
        So, for x = 5 value of polynomial is 0 and hence x = 5 is a zero of polynomial.
(iii)   p(x) = 2x + 5
        p(x) = 0
        2x + 5 = 0
        2x = - 5
         Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
        So, for  Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials value of polynomial is 0 and hence Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials  is a zero of polynomial.
 
(iv)   p(x) = 3x - 2
        p(x) = 0
        3x - 2 = 0
         Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials 
        So, for Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials , value of polynomial is 0 and hence Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials  is a zero of polynomial.
 
(v)    p(x) = 3x
        p(x) = 0
        3x = 0
        x = 0
        So, for x = 0 value of polynomial is 0 and hence x = 0 is a zero of polynomial.
 
(vi)   p(x) = ax
        p(x) = 0
        ax = 0
        x = 0
        So, for x = 0, value of polynomial is 0. Hence x = 0 is a zero of polynomial.
 
(vii)   p(x) = cx + d
        p(x) = 0
        cx+ d = 0
        Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
        So, for Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials, value of polynomial is 0. Hence Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials is a zero of polynomial.
 
Concept Insight: Equate the polynomial to zero and solve the corresponding linear equation to get the value of variable. Be careful while transposing the terms to the other side. For verification substitute the value of the variable obtained in the polynomial.

Chapter 2 - Polynomials Exercise Ex. 2.3

Solution 1
Let p(x) = x3 + 3x2 + 3x + 1.
(i)     x + 1
        Zero of x +1 is-1.
        i.e. p(-1) = (- 1)3 + 3 (- 1)2 + 3 (-1) + 1 = 0
        So, the remainder is 0.
 
(ii)    Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
         Zero of Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials  is Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
         Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
(iii)     x
          Zero of x is 0.
          p(0) = (0)3 + 3(0)2 + 3(0) + 1 = 1
          So, the remainder is 1.
 
(iv)     x + Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
          Zero of x + Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials is:
          x + Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials = 0 Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials x = - Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
          p (- Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials) = (- Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials)3 + 3(- Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials)2 + 3(- Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials) + 1 = - Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials3 + 3 Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials2 - 3Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials + 1
          So, the remainder is - Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials3 + 3 Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials2 - 3Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials + 1
 
(v)      5 + 2x
          Zero of 5+2x  is:
          5 + 2x = 0 Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials 2x = - 5
          i.e. x = - Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
          Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
                                   OR
 
(i)     x + 1
        By long division
 
        Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
         So, remainder is 0.
(ii).    Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials

         By long division

         Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
          So, remainder is Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials .
(iii)     x
          By long division
 
          Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
          So, remainder is 1.
 
(iv)     x + Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
          By long division
 
          Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
         So, the remainder is Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
(v)     5 + 2x 
         By long division
 
         Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
         So the remainder is -Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials.
 
Concept Insight: The remainder of any polynomial p(x) when divided by another polynomial (ax+b) where a and b are real numbers Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials is p(-b/a).
Note that here -b/a  is the zero of polynomial ax+b.
This problem can also be solved using long division. For long division  first write the divisor and dividend in the standard form, i.e. arrange the terms in the descending order of their powers. The process of division is continued till the remainder is constant or the degree of new dividend is less than the degree of divisor. Do not forget to change the sign of terms while subtraction. For cross verification division algorithm
 Dividend = Quotient Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials Divisor + Remainder can be used.

Solution 2
According to the remainder theorem, if p(x) is any polynomial of degree Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials 1 and a is any real number, then when p(x) is divided by the linear polynomial x - a, then the remainder is p(a).

Here p(x) = x3 - ax2 + 6x - a
        p(a) = (a)3 - a(a)2 + 6a - a
               = 5a
So when x3 - ax2 + 6x - a is divided by x - a, remainder comes to 5a.
OR
By long division
 
Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
 
So when x3 - ax2 + 6x - a is divided by x - a, remainder comes to 5a.
 
Concept Insight:  The remainder of any polynomial p(x) when divided by another polynomial (ax+b) where a and b are real numbers Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials is
p(-b/a).
Note that here -b/a  is the zero of polynomial ax+ b.
This question can also be solved using long division method however it is long and time consuming. Chances of making computational error are high in that method.

Solution 3
Zero of 7 + 3x is:
7 + 3x = 0
Therefore, Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
7+3x can be a factor of p(x) = 3x3 + 7x only if  Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
Here p(x) = 3x3 + 7x
 
Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials 7 + 3x is not a factor of 3x3 + 7x.
 
OR
Let us divide (3x3 + 7x) by (7 + 3x). If remainder comes out to be 0 then  7 + 3x will be a factor of
3x3 + 7x.

By long division
 
Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
 
As remainder is not zero so 7 + 3x is not a factor of 3x3 + 7x.

Concept Insight: Any linear polynomial 'ax+b' where a and b are real numbers Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials is a factor of the polynomial p(x) iff p(-b/a) = 0 i.e  -b/a is a zero of p(x) or both the polynomials has a common zero -b/a. This question can also be solved using long division method. Do not forget to change the sign of terms while subtraction in the long division.

Chapter 2 - Polynomials Exercise Ex. 2.4

Solution 1
 
(i)    If (x + 1) is a factor of p(x) = x3 + x2 + x + 1, p (- 1) must be zero.

        Here, p(x) = x3 + x2 + x + 1    
               p(-1) = (- 1)3 + (- 1)2 + (- 1) + 1    
                       = - 1 + 1 - 1 + 1 = 0
        Hence, x + 1 is a factor of this polynomial
(ii)    If (x + 1) is a factor of p(x) = x4 + x3 + x2 + x + 1, p (- 1) must be zero.
        
        Here, p(x) = x4 + x3 + x2 + x + 1    
              p( -1) = (- 1)4 + (- 1)3 + (- 1)2 + (- 1) + 1
                       = 1 - 1 + 1 -1 + 1 = 1
        As, Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
        So, x + 1 is not a factor of this polynomial
 
(iii)    If (x + 1) is a factor of polynomial p(x) = x4 + 3x3 + 3x2 + x + 1, p(- 1) must be 0.

         p(- 1) = (- 1)4 + 3(- 1)3 + 3(- 1)2 + (- 1) + 1
                  = 1 - 3 + 3 - 1 + 1 = 1
         As, Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
         So, x + 1 is not a factor of this polynomial
 
(iv)    If (x + 1) is a factor of polynomial
         p(x) = Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials ,  p(- 1) must be 0.

                Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
         As, Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials

         So, (x + 1) is not a factor of this polynomial.
 
Concept Insight: A linear polynomial 'x-a' is a factor of the polynomial p(x) iff p(a) = 0. Note that 'a' is a zero of polynomials x-a and  p(x) . Be careful while squaring and cubing the numbers.
Solution 2
(i)    If g(x) = x + 1 is a factor of given polynomial p(x), p(- 1)  must be zero.
        p(x) = 2x3 + x2 - 2x - 1
        p(- 1) = 2(- 1)3 + (- 1)2 - 2(- 1) - 1
                 = 2(- 1) + 1 + 2 - 1 = 0
        Hence, g(x) = x + 1 is a factor of given polynomial.
 
(ii)    If g(x) = x + 2 is a factor of given polynomial p(x), p(- 2) must be 0.
        p(x) = x3 +3x2 + 3x + 1
        p(- 2) = (- 2)3 + 3(- 2)2 + 3(- 2) + 1
                 = - 8 + 12 - 6 + 1
                 = - 1
       Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials

        Hence g(x) = x + 2 is not a factor of given polynomial.
 
(iii)    If g(x) = x - 3 is a factor of given polynomial p(x), p(3) must be 0.
            p(x) = x3 - 4 x2 + x + 6
            p(3) = (3)3 - 4(3)2 + 3 + 6
                   = 27 - 36 + 9 = 0
        So, g(x) = x - 3 is a factor of given polynomial.
 
Concept Insight: The problem is a direct application of Factor theorem. g(x) will be the factor of the polynomial p(x) iff the zero of the linear polynomial g(x) when put in place of the variable of polynomial results to zero. Be careful while squaring and cubing the numbers.
Solution 3
If x - 1 is a factor of polynomial p(x), then p(1) = 0
 
(i)    p(x) = x2 + x + k  
       p(1) = 0
       Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials (1)2 + 1 + k = 0
       Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials 2 + k = 0
       Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials k = - 2
        So, value of k is - 2.
 
Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
Concept Insight: x-1 is a factor of the given polynomial p(x) iff p(1) = 0 thus equating p(1) to zero will give the required value of constant k. Be careful with arithmetic simplifications.
 
Solution 4
(i)    12x2 - 7x + 1    
       The two numbers such that pq = 12 Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials 1 = 12 and p + q = - 7. They are p = - 4 and
       q = - 3
       Now, 12x2 - 7x + 1 = 12x2 - 4x - 3x + 1    
                                   = 4x (3x - 1) - 1 (3x - 1)
                                   = (3x - 1) (4x - 1)
 
(ii)    2x2 + 7x + 3
        The two numbers such that pq = 2 Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials 3 = 6 and p + q = 7.
        They are p = 6 and q = 1
        Now, 2x2 + 7x + 3 = 2x2 + 6x + x + 3    
                             = 2x (x + 3) + 1 (x + 3)
                             = (x + 3) (2x+ 1)
 
(iii)    6x2 + 5x - 6    
        The two numbers such that pq = - 36 and p + q = 5.
        They are p = 9 and q = - 4
        Now,
        6x2 + 5x - 6 = 6x2 + 9x - 4x - 6    
                    = 3x (2x + 3) - 2 (2x + 3)
                    = (2x + 3) (3x - 2)

(iv)    3x2 - x - 4    
         The two numbers such that pq = 3 Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials (- 4) = - 12
         and p + q = - 1.
         They are p = - 4 and q = 3.
         Now,
         3x2 - x - 4 = 3x2 - 4x + 3x - 4    
                  = x (3x - 4) + 1 (3x - 4)
                  = (3x - 4) (x + 1)

Concept Insight: To factorise the polynomial ax2+bx+c, by splitting the middle term, 
b is expressed as the sum of two numbers whose product is ac.
Do not forget to consider the sign of the terms while splitting.
Remember
 

ac>0

b>0

  b =(p+q) where p>0,q>0

ac>0

b<0

  b =(p+q) where p<0,q<0

ac<0

b>0

  b =(p+q) where
p > q then p>0 and q<0

ac<0

b<0

b =(p+q) where
p > q then p<0 and q>0

Solution 5
 
(i)    Let p(x) = x3 - 2x2 - x + 2
       Factors of 2 are 1, 2.
       By hit and trial method
       p(2) = (2)3 - 2(2)2 - 2 + 2
           = 8 - 8 - 2 + 2 = 0
       So, (x - 2) is factor of polynomial p(x)
   
       By long division
       Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
       Now,     Dividend = Divisor Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials Quotient + Remainder
    Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials x3 - 2x2 - x + 2 = (x + 1) (x2 - 3x + 2) + 0
                  = (x + 1) [x2 - 2x - x + 2]
                  = (x + 1) [x (x - 2) - 1 (x - 2)]
                  = (x + 1) (x - 1) (x - 2)
                  = (x - 2) (x - 1) (x + 1)
 
(ii)    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
 
       Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
        Now, Dividend = Divisor Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials Quotient + Remainder
        Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials 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)
 
(iii)    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
 
         Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
         We know that
         Dividend = Divisor Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials 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)
 
(iv)    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,
         Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
          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)

 
Concept Insight: To factorise p(x) when its degree is greater than or equal to 3 note down all the factors of constant term considering both negative and positive sign.
Check the obtained factors for the possible zeroes of the polynomial p(x)  Using Factor theorem one zero can be obtained continue the process till all the zeroes are obtained or use long division method. To obtain the other quadratic factor use long division to determine the other factors. The degree of the polynomial is  less than or  equal to the number of real factors the polynomial.

Chapter 2 - Polynomials Exercise Ex. 2.5

Solution 1
(i).    By using identity Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
        Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
(ii).    By using identity Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
         Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
(iii).    Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
 
           By using the identity  Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
           Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
(iv).    By using identity Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 

          Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
(v).    By using identity Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
         Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
 
Concept Insight: If the value of the two terms of the binomials are equal then use the algebraic identity (x+a) (x-a) = x2 - a2 else use (x+a) (x+b) = x2+(a+b)x+ab to obtain required product. 
Solution 2
Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
Concept Insight: Reduce the terms of the polynomial to perfect cube and then if the two terms of the polynomial are separated by positive sign use the identity Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials  and when by negative sign use Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials .
Solution 3
We Know that
Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials        
Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
Concept Insight: Reduce each terms of the polynomial as per the left hand side of the standard identity,  Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials .
Solution 4
We know that
Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
                         Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
Concept Insight: Since the left hand side of the identity resembles the left hand side of identity, Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials , so this identity
 
will be applicable here. Now the right hand side of the above identity can be written into many forms we need to look at what is required to proved, Accordingly apply  mathematical simplifications and square identities to get the desired result.
Solution 5
Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
Concept Insight: Use the result that  Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials  for x + y + z = 0.
Solution 6
Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
Concept Insight: Use the result Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials  since x + y + z = 0. Also consider the
 
sign of the term. Carefully do the computation.
Solution 7
We know that,
 Area = length Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials breadth
Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
Concept Insight: For such questions factorise the expression, given for the area of rectangle by splitting the middle term. One of its factors will be its length and the other will be its breadth.
Solution 8
We know that,
Volume of cuboid = length Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials breadth Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials height
(i).    Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials

        So, the possible solutions is
        Length = 3, breadth = x, height = x - 4
 
        Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
Concept Insight: For such questions factorise the expression, given for the volume of the cuboid by taking the common term out if it has two terms and by splitting the middle term if the polynomial has three terms. Three factors obtained will be its length  breadth and  height.
Solution 9
(i).    103 Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials 107 = (100 + 3) (100 + 7)
                          = (100)2 + (3 + 7) 100 + (3) (7)
        [By using the identity, Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials, where

        x = 100, a = 3 and b = 7]
                          = 10000 + 1000 + 21
                          = 11021
(ii).   95 Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials 96 = (100 - 5) (100 - 4)
                    = (100)2 + (- 5 - 4) 100 + (- 5) (- 4)
        [By using the identity, Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials, where

        x = 100, a = - 5 and b = - 4]
                    = 10000 - 900 + 20
                    = 9120
(iii).  104 Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials 96 = (100 + 4) (100 - 4)
                    = (100)2 - (4)2           Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials


                    = 10000 - 16
                    = 9984
 
Concept Insight: The key is to use the algebraic identity (x+a) (x+b) = x2+(a+b)x+ab or (x+a) (x-a) = x2 - a2  for such questions. Write each of the numeral as  100 k ,  or any other suitable number whose square can be easily computed.
Solution 10
Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
Concept Insight: Use the appropriate square identity.  If the polynomial has only two terms, reduce each term to the perfect square and use the algebraic identityNcert Solutions Cbse Class 9 Mathematics Chapter - Polynomials . When the polynomial has three terms and the term having
 
unit power of each variable has negative sign use the square identity Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials else use Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials .
Solution 11
Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
 
Concept Insight: Use the algebraic identity Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials 
 
. Do consider the sign of terms while multiplying and squaring.
Solution 12
Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
Concept Insight: Use the algebraic identity  Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
  in the reverse order. Write each term as per the terms of the standard identity. Do consider the sign of terms involved.
Solution 13
Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
Concept Insight: Since the expressions involves cube so cubic identity will be used.  If the terms of the given polynomial are separated by positive sign use the identity  Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials or if  negative signs are used  then use Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials. Carefully apply the mathematical operations.
Solution 14
We know that
 
Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
(i)    (99)3 = (100 - 1)3
                = (100)3 - (1)3 - 3(100) (1) (100 - 1)
                = 1000000 - 1 - 300(99)
                = 1000000 - 1 - 29700
                = 970299

(ii)    (102)3 = (100 + 2)3
                  = (100)3 + (2)3 + 3(100) (2) (100 + 2)
                  = 1000000 + 8 + 600 (102)
                  = 1000000 + 8 + 61200
                  = 1061208


(iii)    (998)3 = (1000 - 2)3
                   = (1000)3 - (2)3 - 3(1000) (2) (1000 - 2)
                   = 1000000000 - 8 - 6000(998)
                   = 1000000000 - 8 - 5988000
                   = 1000000000 - 5988008
                   = 994011992
 
 
Concept Insight: Use the cubic identity Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials and Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials . Write the numerical term as something added or
 
subtracted from 10,100, 1000 or higher powers of 10 as it's easy to compute higher powers of 10. Carefully apply the mathematical operations.
Solution 15
 
Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
Concept Insight: Since all the polynomial given here have degree 3 so cubic identities would be used here. Now if all the terms of the given polynomial are positive  then use identity  Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials while if any two terms has negative sign reduce each of
 
the term of the polynomial as per the standard cubic identity Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
Solution 16
Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials
 
Concept Insight: When the two terms of the polynomial are separated by positive sign use the identity Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials  and when by negative sign use Ncert Solutions Cbse Class 9 Mathematics Chapter - Polynomials.
 
Carefully take the common term out.