Question

Find all the zeroes of the polynomial x³+3x²-2x-6, If two of its zeroes are -√2 and √2.

Answer 1

Here is the solution :

The shortest method possible :

In general, the sum of all the roots of a cubic polynomial "ax³ + bx² + cx + d =0" is -b/a and Product of the roots is -d/a,

Now, 
Given that,

Cubic Polynomial : x³ + 3x² -2x -6,
roots : -√2 , +√2, and let another one be x,

=> From the above statement, 
-√2 + √2 + x = -(3)/1 = -3,

=> x = -3,

Also, we know that, Cubic polynomial has at most 3 zeroes or roots !

Here we got another root as -3,

=> the roots must be -√2 , +√2 , -3,

Proof :

Product must be equal to -(-6)/1 = 6,
=> -√2 * +√2 * -3 must be 6,
=> -2 * -3 = +6,

Therefore the root we got is correct !

Therefore : All the zeroes of the given cubic polynomial are -√2, +√2, -3 !

Hope you understand, Have a Great Day !
Thanking you, Bunti 360 !!