Question

examine weather x+1 and 2x-3 are factor of 2x^3-9x^2+x+12

Answer 1

given, p(x) = 2x^3 - 9x^2 + x + 12 ---eq(1)

let ,g(x) =( x + 1 ), ( 2x - 3) =0

x = - 1 , x = 3/2

check ,at x = -1 in eq (1)

p(-1)= 2(-1)^3 - 9(-1)^2 + (-1)+ 12

p(-1) = - 2 - 9 - 1 + 12

p(-1) = -12 + 12 = 0

p(-1) = 0

hence, ( x + 1 ) is a factor of p(x)

now check, at x = 3/2 in eq(1) , we get

p(3/2) =2(3/2)^3 - 9(3/2)^2 + 3/2 + 12

p(3/2) = 54/8 - 81/4 + 3/2 + 12

p(3/2) =( 27/4 + 3/2 ) + 12 - 81/4

p(3/2) = (33/4 + 12 ) - 81/4

p(3/2)= 81/4 - 81/4

p(3/2) = 0

hence, ( 2x - 3 ) is also a factor of p(x).

therefore,( x + 1 ), ( 2x - 3 ) are factors of given polynomial p(x).