10th Maths 2.3

NCERT Class 10th solution of Exercise 2.2

NCERT Class 10th solution of Exercise 2.1

NCERT Class 10th solution of Exercise 3.1

NCERT Class 10th Maths Projects

Exercise 2.3

Q1. Divide the polynomial `p(x)` by the polynomial `g(x)` and find the quotient and remainder in each of the following :

i) `p(x)=x^3-3x^2+5x-3, g(x)=x^2-2`

ii) `p(x)=x^4-3x^2+4x+5, g(x)=x^2+1-x`

iii) `p(x)=x^4-5x+6, g(x)=2-x^2`

Q2. Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial :

i) `t^2-3, 2t^4+3t^3-2t^2-9t-12`

ii) `x^2+3x+1, 3x^4+5x^3-7x^2+2x+2`

iii) `x^3-3x+1, x^5-4x^3+x^2+3x+1`

Q3. Obtain all other zeroes of `3x^4+6x^3-2x^2-10x-5`, if two of its zeroes are `sqrt(5/3)` and              `-sqrt(5/3)`.

Sol. 

 `sqrt(5/3)` and `-sqrt(5/3)` are two zeroes of the given polynomial. So, 

`=(x + sqrt(5/3))(x - sqrt(5/3))`

`=(x^2 - 5/3)`

`=(3x^2 - 5)`

Q4. On dividing `x^3-3x^2+x+2` by a polynomial `g(x)`, the quotient and remainder were `x-2` and `-2x+4`, respectively. Find `g(x)`.

Sol. 

By using Euclid's Division Algorithm

`g(x)(x-2)+(-2x+4)=x^3-3x^2+x+2`

`g(x)(x-2)=x^3-3x^2+x+2+2x-4`

`g(x)=(x^3-3x^2+3x-2)/(x-2)`

Q5. Give examples of polynomials `p(x),g(x),q(x)` and `r(x)` which satisfy the division algorithm and  i) deg`p(x)`=deg`q(x)` ii) deg `q(x)`=deg`r(x)` iii) deg `r(x)=0`.

Answer

i) `p(x)=2x^2-2x+14, g(x)=2,`

`q(x)=x^2-x+7, r(x)=0`

ii) `p(x)=x^3+x^2+x+1, g(x)=x^2-1,`

`q(x)=x+1, r(x)=2x + 2`

iii) `p(x) = x^3 + 2x^2 - x + 2, g(x)=(x^2-1),`

`q(x)=x+2,  r(x)=4`

There can be several example in each of (i), (ii), and (iii).