 Hi, and welcome to the session. My name is Reshi. I'm going to help you to solve the following question. Question is, obtain all other zeros of 3x raised to the power 4 plus 6x cube minus 2x square minus 10x minus 5 if two of its zeros are root 5 upon 3 and minus root 5 upon 3. First of all, we should understand that a real number alpha is a zero of a polynomial fx if f alpha is equal to zero. This is the key idea for solving the given question. Let us now start with the solution. Let fx is equal to 3x raised to the power 4 plus 6x cube minus 2x square minus 10x minus 5 and its zeros are root 5 upon 3 and minus root 5 upon 3 implies x minus root 5 upon 3 multiplied by x plus root 5 upon 3 are factors of fx which can be further written as x square minus 5 upon 3 which is equal to 1 upon 3 multiplied by 3x square minus 5. We can recall by factor theorem that x minus root 5 upon 3 and x plus root 5 upon 3 are the factors of fx. This implies 3x square minus 5 is a factor of fx. Now we divide fx by 3x square minus 5. Now we will start the division. Since the first term of the dividend is 3x raised to the power 4 so we will multiply the divisor with x square to get the desired term. Multiplying the divisor with x square we get 3x raised to the power 4 minus 5x square. Now subtracting the light terms and bringing down the rest of the terms of the dividend we get 6x cube plus 3x square minus 10x minus 5. Now first term is 6x cube so we will multiply the divisor with 2x. Multiplying the divisor with 2x we get 6x cube minus 10x. Now subtracting the light terms and bringing down the rest of the terms we get 3x square minus 5. Now to get 3x square we will multiply the divisor with 1 so now the next term of the quotient would be 1. Subtracting the light terms we get remainder equal to 0. We know dividend is equal to divisor multiplied by quotient plus remainder. Now substituting these values we get 3x raised to the power 4 plus 6x cube minus 2x square minus 10x minus 5 which is our dividend is equal to 3x square minus 5 multiplied by x square plus 2x plus 1 plus 0. 0 is our remainder 3x square minus 5 is our divisor and x square plus 2x plus 1 is the quotient obtained in the subdivision can be further written as root 3x square minus root 5 square and x square plus 2x plus 1 is equal to x plus 1 whole square. Here we have used the identity that a plus v whole square is equal to a square plus b square plus 2ab. This is further equal to root 3x plus root 5 multiplied by root 3x minus root 5 multiplied by x plus 1 whole square. We have applied the formula of a square minus b square is equal to a plus b into a minus b. Now for finding the zeros of fx we will put all the vectors of fx equal to 0. Therefore root 3x plus root 5 is equal to 0 which implies x is equal to minus root 5 upon root 3 which is further equal to minus root 5 upon 3. Similarly root 3x minus root 5 is equal to 0 which implies x is equal to under root of 5 upon 3. Similarly we can write x plus 1 multiplied by x plus 1 is equal to 0 which implies x is equal to minus 1 or x is equal to minus 1. So the first zeros of the given fx is equal to minus root 5 upon 3 root 5 upon 3 minus 1 and minus 1. Hence four zeros given fx are minus 1 minus 1 root 5 upon 3 and minus root 5 upon 3. This is our required answer. This completes the session. Hope you enjoyed the session. Goodbye.