Polynomials| Worksheet Solutions| Class 10

Worksheet solutions for chapter: Polynomials for class 10 are provided below.

 

(1) The graphs of y = p(x) are given in figure given below, for some polynomials p(x). Find the number of zeroes of p(x), in each case.

(i)

(ii)

Ans-

(i) The graph cuts x – axis at two points.

Therefore, the graph has two zeroes.

(ii)The graph do not cut  x – axis at any point.

Therefore, the graph has no zeroes.

 

(2) Find a quadratic polynomial, the sum and product of whose zeroes are -5 and -10 respectively.

Ans –

(3) Divide x2 + 2x + 4 by polynomial x+1 and find quotient and remainder.

Ans-

Dividing x2 + 2x + 4 by polynomial x+1

Therefore, quotient = x + 1 and remainder = 3

Hence, quotient = x + 1 and remainder = 3

(4) Find the zeroes of polynomial x2 + 10x + 25. Verify relationship between zeroes and the coefficients.

Ans –

(i) x2 + 10x + 25

= x2 + 5x + 5x + 25

=x(x + 5) +5(x + 5)

=(x + 5)(x + 5)

Therefore, the value of zeroes of polynomial x2 +10x + 25 is

x + 5 = 0, ie, x = -5

x + 5 + 0, ie, x = -5

The zeroes of polynomial x2 + 10x + 25 are -5 and – 5.

Now,

(5) Find the zeroes of polynomial 2x2 -5x + 2.

Ans-

2x2 -5x + 2

2x2 -4x – x + 2

2x(x – 2)- 1(x – 2)

(x – 2) (2x – 1)

Therefore, the value of zeroes of polynomial 2x2 – 5x + 2 is

x – 2 = 0, ie, x = 2

2x – 1 = 0, ie, x = 1/2

The zeroes of polynomial 2x2 – 5x + 2 are 2 and 1/2.

 

(6) On dividing x3 – 2x2 + 2x + 2 by a polynomial g(x), the quotient and remainder were x- 4 and 2x + 6 respectively.

Ans –

Dividend = x3 – 2x2 + 2x + 2

Divisor = g(x)

Quotient = x – 4

Remainder = 2x + 6

Dividend = Divisor  Quotient + Remainder

x3 – 2x2 + 2x + 2 = g(x)  (x – 4) + (2x + 6)

x3 – 2x2 + 2x + 2 – (2x + 6) = g(x)  (x – 4)

x3 – 2x2 + 2x + 2 – 2x – 6 = g(x)  (x – 4)

x3 – 2x2 -4 = g(x)  (x – 4)

Hence, g(x) is x2 +2x + 8.

 

(7) Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial:

x -2,  x3 + 2x2 – 4x + 8

Ans-

Division of the second polynomial by the first polynomial is showing below-

Remainder is zero.

Hence, the first polynomial (x – 2) is a factor of the second polynomial.

 

(8) Find all other zeroes of the of x3 + x2  + 2x + 2 , if one zero is -1.

Ans-

Ans- Since x = -1 is a zero of the x3 + x2 + 2x + 2.

Therefore, x + 1 is a factor of x3 + x2 + 2x + 2.

So, x3 + x2 + 2x + 2= (x + 1) (x2 + 2)

Now,

x + 1 = 0, ie, x= -1

x2 + 2 = 0,ie, x = ± √2

Hence, x3 + x2 + 2x + 2 has three zeroes, ie -1, √2 and – √2.

 

(9) Divide x2 + 10 x + 16 by x + 3

Ans-

Divide x2 + 10 x + 16 by x + 3

Therefore, quotient = x + 7

Remainder = 7

 

(10) State whether statement is ‘True’ or ‘False’.

(i) A quadratic polynomial can have at most 2 zeroes and a cubic polynomial can have at most 3 zeroes.

(ii) If  are the zeroes of te cubic polynomial ax3 + bx2 + cx + d, then

α+ β + γ = c/a

(iii) If p(x) and g(x) are any to polynomials with g(x)  0, then we can find polynomials q(x) and r(x) such that

p(x) = g(x)  r(x) + q(x),

where r(x) = 0 or degree of r(x) < degree of g(x).

Ans-

(i) A quadratic polynomial can havt at most 2 zeroes and a cubic polynomial can have at most 3 zeroes.   ‘True’

(ii) If  are the zeroes of te cubic polynomial ax3 + bx2 + cx + d, then

+  +  = c/a      ‘False’

(iii) If p(x) and g(x) are any to polynomials with g(x)  0, then we can find polynomials q(x) and r(x) such that

p(x) = g(x)  r(x) + q(x),

where r(x) = 0 or degree of r(x) < degree of g(x).   ‘False’

 

Helping Topics

Polynomials

NCERT Solutions Class 10