Problem 1 :
Find all the zeroes of the polynomial x^{3} + 3x^{2} – 2x – 6, if two of its zeroes are -√2 and √2.
Solution :
Given, polynomial x^{3} + 3x^{2} – 2x – 6
Two of its zeroes are -√2 and √2
(x – (-√2)) and (x – (√2))
(x + √2) (x - √2)
x^{2} - x√2 + x√2 – 2
x^{2} – 2
x + 3 = 0
x = -3
All the zeroes are -√2, √2 and -3
Problem 2 :
Find all the zeroes of the polynomial 2x^{3} + x^{2} – 6x – 3, if two of its zeroes are -√3 and √3.
Solution :
Given, polynomial 2x^{3} + x^{2} – 6x – 3
Two of its zeroes are -√3 and √3
(x – (-√3)) and (x – (√3))
(x + √3) (x - √3)
x^{2} - x√3 + x√3 – 3
x^{2} – 3
2x + 1 = 0
2x = -1
x = -1/2
All the zeroes are -√3, √3 and -1/2.
Problem 3 :
Obtain all other zeroes of the polynomial 2x^{3} - 4x – x^{2} + 2, if two of its zeroes are √2 and -√2.
Solution :
Given, polynomial 2x^{3} – x^{2} – 4x + 2
Two of its zeroes are -√2 and -√2
(x – (√2)) and (x – (-√2))
(x - √2) (x + √2)
x^{2} + x√2 - x√2 – 2
x^{2} – 2
2x - 1 = 0
2x = 1
x = 1/2
All the zeroes are - √2, -√2 and 1/2.
Problem 4 :
If the polynomial 6x^{4} + 8x^{3} + 17x^{2} + 21x + 7 is divided by another polynomial 3x^{2} + 4x + 1 then the remainder comes out to be ax + b, find ‘a’ and ‘b’.
Solution :
Given, polynomial 6x^{4} + 8x^{3} + 17x^{2} + 21x + 7
Divided by another polynomial 3x^{2} + 4x + 1
Remainder comes out to be ax + b,
x + 2
To find ‘a’ and ‘b’ :
a = 1, b = 2
2x + 1 = 0
2x = -1
x = -1/2
All the zeroes are -√3, √3 and -1/2.
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM