obtain all other zeroes of 3(x^4)+6(x^3)-2(x^2)-10x-5 if two of its zeroes are sqrt(5/3) and [sqrt(5/3)]

ncert cbse 10th chapter 2  exercise 2.3

3. obtain all other zeroes of 3(x^4)+6(x^3)-2(x^2)-10x-5 if two of its zeroes are sqrt(5/3) and [sqrt(5/3)]

factors are

[x-sqrt(5/3)][x-sqrt(5/3)] = [(x^2) - (5/3)]=(1/3) [3(x^2) -5]

now divide the given expression 3(x^4)+6(x^3)-2(x^2)-10x-5 with [3(x^2) -5]

using long division

                           (x^2) +2x +1                                

                       ---------------------------------------------------------------------------

3(x^2)+0x -5 |  3(x^4)+6(x^3)- 2(x^2) - 10x  -5 

                      |  3(x^4)+0(x^3) -5(x^2)

                      -------------------------------------------------------------------

                     |              6(x^3) +3(x^2)- 10x 

                     |              6(x^3) +0(x^2) - 10x

                      -----------------------------------------------------------------

                    |                            3(x^2) +0x -5

                    |                             3(x^2) +0x -5

                    -------------------------------------------------------------

                   |                                        0   

 

 

 

 

 

 

 

 

we get the quotient as  (x^2) +2x +1      

on factoring we get (x+1)(x+1) whose zeroes are (-1) and (-1)

 

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

 

let p(x)  = (x^3) – 3(x^2) + x + 2

quotient =q(x)  = (x-2)

remainder r(x) = (-2x+4)

p(x) = g(x)*q(x) +r(x)

g(x) = [p(x) - r(x)] / [q(x)]

g(x) = [{(x^3) – 3(x^2) + x + 2} - {(-2x+4)}] / [(x-2)]

g(x) = [( x^3) – 3(x^2) +3x – 2 ] / [x-2]

using long division

g(x) =  (x^2) -1x+1

 

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  ncert cbse 10th  mathematics chapter 2 optional exercise
 If the zeroes of the polynomial (x^3) – 3(x^2) + x + 1 are a – b, a, a + b, find a and b.
solution
 
2. Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2, –7, –14 respectively
solution



4. If two zeroes of the polynomial (x^4) – 6(x^3) – 26(x^2) + 138x – 35 are 
 [2 ±sqrt(3) ] , find other zeroes

 solution
 
5. If the polynomial (x^4) – 6(x^3) + 16(x^2) – 25x + 10 is divided by another polynomial (x^2) – 2x + k, the remainder comes out to be x + a, find k and a
solution  
 
exercise 2.3
 

3. obtain all other zeroes of 3(x^4)+6(x^3)-2(x^2)-10x-5 if two of its zeroes are sqrt(5/3) and [sqrt(5/3)]
solution
 
4. On dividing (x^3) – 3(x^2) + x + 2 by a polynomial g(x), the quotient and remainder were x – 2
and –2x + 4, respectively. Find g(x).   
solution
 
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