Find the value of k if x2+2x+k is a factor of 2x4+14x2+5x+6Also find all the zeroes of polynomial

Find the value of k if x2+2x+k is a factor of 2x4+14x2+5x+6Also find a

1.	Find the value of k if x2+2x+k is a factor of 2x4+14x2+5x+6Also find all the zeroes of polynomial
Answer» If g(x) = x2 + 2x + k is a factor of f(x) = 2x4 + x3 - 14x2 + 5x + 6, then remainder is zero when f(x) is divided by g(x).Let quotient = Q and remainder = RLet us now divide f(x) by g(x).R = x(7k + 21) + (2k2 + 8k + 6) -------(1) and Q = 2x2 - 3x - 2(k + 4).------------(2)Now, R = 0.{tex}\\Rightarrow{/tex}\xa0x (7k + 21) + 2 (k2 + 4k + 3) = 0\xa0{tex}\\Rightarrow{/tex}\xa07x (k + 3) + 2 (k+1)(k+3) = 0{tex}\\Rightarrow{/tex}\xa0(k+3) [7x + 2(k+1)] = 0{tex}\\Rightarrow{/tex}\xa0k + 3 = 0{tex}\\Rightarrow{/tex}\xa0k = -3Thus, polynomial f(x) can be written as,2x4 + x3 - 14x2 + 5x + 6 = (x2 + 2x + k) [2x2 - 3x - 2(k + 4)] = (x2 + 2x - 3) (2x2 - 3x - 2)Zeros of\xa0x2 + 2x - 3 are,x2 + 2x - 3 = 0{tex}\\Rightarrow{/tex}\xa0(x + 3) (x - 1) = 0{tex}\\Rightarrow{/tex}\xa0x = -3 or x = 1Zeros of\xa0(2x2 - 3x - 2) are,2x2 - 3x - 2 = 0{tex}\\Rightarrow{/tex}\xa02x2 - 4x + x - 2 = 0{tex}\\Rightarrow{/tex}\xa02x(x - 2) + 1(x - 2) = 0{tex}\\Rightarrow{/tex}\xa0(x - 2)(2x + 1) = 0x = 2 or x = -{tex}\\frac12{/tex}Thus, the zeros of f(x) are: -3 ,1, 2 and\xa0-{tex}\\frac12{/tex}