Question

check whether p(x) is a multiple of g(x) or not p(x)= 2xclub -11xsquare -4x +5 ,g(x)=2x+1

Answer 1

p(x)=2x^3-11x^2-4x+5
g(x)=2x+1
        2x+1=0
         x=-1/2
p(-1/2)=2(-1/2)^3-11(-1/2)^2-4(-1/2)+5
           = -1/4-11/4+2+5
           =-12/4+7
           =-3+7
           =4 

Answer 2

$p(x) = 2x {}^{3} - 11x {}^{2} - 4x + 5 \\ g(x) = 2x + 1 \\ by \: factor \: theorem \\ g(x) = 0 \\ 2x + 1 = 0 \\ x = \frac{ - 1}{2} \\ remainder = p( \frac{ - 1}{2} ) \\ = 2 \times ( \frac{ - 1}{2} ) {}^{3} - 11 \times (\frac{ - 1}{2} ) {}^{2} - 4 \times \frac{ - 1}{2} + 5 \\ = 2 \times \frac{ - 1}{8} - 11 \times \frac{1}{4} + \frac{4}{2} + 5 \\ = \frac{ - 1}{4} - \frac{11}{4} + 2 + 5 \\ = \frac{ - 1 - 11 + 28}{4} \\ = \frac{16}{4} = 4$
So, p(x)is not a multiple of g(x).