The quotient of x5 – 3x3 – 3x2 – 10x + 15 and a polynomial is x2 – 5. What is the polynomial? x5 – 3x3 – 2x2 – 10x + 10 x7 – 8x5 – 3x4 + 5x3 + 30x2 + 50x – 75 x3 + 2x – 3 x5 – 3x3 – 4x2 – 10x + 20

Answer:\(x^{}7{}-8x^{}5{} - 3x^{}4{} +5 x^{}3{}+30x^{}2{}+50x-75\\\)If the quotient of the polynomial x⁵ – 3x³ – 3x² – 10x + 15 is given to be polynomial (x² – 5). The unknown polynomial (dividend)will be the product of the quotient given and its divisor. Dividend = Divisor * quotientDividend is calculated as shown; \((x^{}5{}-3x^{}3{} -3x^{}2{} - 10x + 15)(x^{}2{} - 5)\\= x^{}7{}-5x^{}5{} -3x^{}5{} + 15 x^{}3{} - 3x^{}4{} + 15x^{}2{} -10x^{}3{}+50x+15x^{}2{} -75\\= x^{}7{}-8x^{}5{} - 3x^{}4{} +15 x^{}3{}-10 x^{}3{}+ 15x^{}2{}+ 15x^{}2{}+50x-75\\= x^{}7{}-8x^{}5{} - 3x^{}4{} +5 x^{}3{}+30x^{}2{}+50x-75\\\)The final expression gives the requi polynomial.