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WORKSHEET 1 - Polynomials

1.  The graphs of y = p (x) are given in the figure below for some polynomials p(x). Find the number of zeroes of p (x).
  • Solution:no zeroes.

 2.  The graphs of y = p (x) are given in the figure below for some polynomials p(x). Find the number of zeroes of p (x).
  • Solution:One

 3.  The graphs of y = p (x) are given in the figure below for some polynomials p(x). Find the number of zeroes of p (x).
  • Solution:Three

 4.  The graphs of y = p (x) are given in the figure below for some polynomials p(x). Find the number of zeroes of p (x).
  • Solution:Two

 5.  The graphs of y = p (x) are given in the figure below for some polynomials p(x). Find the number of zeroes of p (x).
  • Solution:Four

 6.  The graphs of y = p (x) are given in the figure below for some polynomials p(x). Find the number of zeroes of p (x).
  • Solution:Three.

 7.  Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
(i) x2 – 2x – 8 (ii) 4s2 – 4s + 1 (iii) 6x2 – 3 – 7x
(iv) 4u2 + 8u (v) t2 – 15 (vi) 3x2 – x – 4.
  • Solution:(i) x2 – 2x – 8
    = x2 – 4x + 2x – 8
    = x(x – 4) + 2(x – 4)
    = (x – 4)(x + 2)
    Therefore the zeroes of the polynomial x2 – 2x – 8 are {4, -2}.
    Relationship between the zeroes and the coefficients of the polynomial:
    Sum of the zeroes = - 
    = -
    Also sum of the zeroes of the polynomial = 4 – 2 = 2.
    Product of the zeroes = =
    Also product of the zeroes = 4 x –2 = -8
    Hence verified.
    (ii) 4s2 – 4s + 1
    = 4s2 – 2s – 2s + 1
    = 2s(2s – 1) – 1(2s – 1)
    = (2s – 1)(2s – 1)
    = 2(s - 2(s - 
    Therefore the zeroes of the polynomial are 
    Relationship between the zeroes and the coefficients of the polynomial:
    Sum of the zeroes = -  = -  = 1
    Also sum of the zeroes = = 1
    Product of the zeroes = =
    Also product of the zeroes = 
    Hence verified.
    (iii) 6x2 – 3 – 7x
    = 6x2 – 7x – 3
    = 6x2 – 9x + 2x – 3
    = 3x(2x – 3) + 1(2x – 3)
    = (2x –3)(3x + 1)
    = 2(x - 3(x + 
    = 6(x - (x + 
    The zeroes of the polynomials are {

    Relationship between the zeroes and the coefficients of the polynomial:
    Sum of the zeroes = -  = -  = 
    Also sum of the zeroes = 
    Product of the zeroes = =
    Also product of the zeroes = 
    Hence verified.
    (iv) 4u2 + 8u
    = 4u(u + 2)
    = 4[u – 0][u –(- 2)]
    The zeroes of the polynomials are {0, -2}
    Relationship between the zeroes and the coefficients of the polynomial:
    Sum of the zeroes = -  = -  
    Also sum of the zeroes = 
    Product of the zeroes = =
    Also product of the zeroes = 
    Hence verified.
    (v) t2 – 15
    = (t + 
    The zeroes of the polynomials are {
    Relationship between the zeroes and the coefficients of the polynomial:
    Sum of the zeroes = -  = -  
    Also sum of the zeroes = 
    Product of the zeroes = =
    Also product of the zeroes = 
    Hence verified.
    (vi) 3x2 – x – 4.
    = 3x2 – 4x + 3x – 4
    = x(3x – 4) + 1(3x – 4)
    = (3x – 4)(x + 1)
    The zeroes of the polynomials are {
    Relationship between the zeroes and the coefficients of the polynomial:
    Sum of the zeroes = -  = -  
    Also sum of the zeroes = 
    Product of the zeroes = =
    Also product of the zeroes = 
    Hence verified.

 8.  Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
  • Solution:(i) 
    Let the quadratic polynomial be ax2 + bx + c, and its zeroes be a and b
    Given a + b =  = 
    a b = -1 = 
    If a = 4 , b = -1 and c = -4
    The quadratic polynomial is 4x2 - x - 4.
    (ii) 
    Let the quadratic polynomial be ax2 + bx + c, and its zeroes be a and b
    Given a + b =  = 
    a b =  = 
    If a = 1, b = - and c = 
    The quadratic polynomial x2 - x + (or) 3x2 - 3x + 1 .
    (iii) 0, 
    Let the quadratic polynomial be ax2 + bx + c and the zeroes be a + b
    Given a + b = 0 = 
    a b =  = 
    If a = 1, b = 0 and c = 
    The quadratic polynomial is x2 + 
    (iv) 1, 1
    Let the quadratic polynomial be ax2 + bx + c , and its zeroes be a + b
    Given a + b =  = 1
    a b =  = 1
    \ If a = 1 , b = -1 and c = 1
    \ The quadratic polynomial is x2 - x + 1.
    (v)  , 
    Let the quadratic polynomial be ax2 + bx + c , and its zeroes be a + b
    Given a + b = = 
    a b =.
    If a = 4, b = 1 and c = 1
    The quadratic polynomial is 4x2 + x + 1.
    (vi) 4, 1
    Let the quadratic polynomial be ax2 + bx + c
    Given a + b = = 4
    a b = = 1
    If a = 1, b = -4 and c = 1
    \ The quadratic polynomial is x2 – 4x +1.

 9.  Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following:
(i) p(x) = x3 – 3x2 + 5x + 3, g(x) = x2 – 2
(ii) p(x) = x4 – 3x2 + 4x + 5, g(x) = x2 + 1 – x
  • Solution:(i) p(x) = x3 – 3x2 + 5x + 3, g(x) = x2 – 2



    Quotient is (x - 3)
    Remainder is 7x – 9
    (ii) p(x) = x4 – 3x2 + 4x + 5, g(x) = x2 + 1 – x
    Rearrange g(x) as x2 - x + 1

    The Quotient is x2 + x – 3
    Remainder is 8
    (iii) p(x) = x4 – 5x + 6, g(x) = 2 – x2
    Rearrange g(x) as –x2 + 2
    Quotient is –x2 – 2
    Remainder is –5x + 10

 10.  Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial.
(i) t2 – 3, 2t4 + 3t3 – 2t2 – 9t – 12
(ii) x2 + 3x +1, 3x4 + 5x3 – 7x2 + 2x + 2
(iii) x3 – 3x + 1, x5 – 4x3 + x2 + 3x + 1 
 

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