Question

If x+1 is a factor of 2x³+ax^{​​​​​2}+2bx+1, then find the value of a and b given that 2a-3b=4.

Answer 1

given that (x+1) is a factor of p(x)

therefore, -1 is a zero of given p(x)

p(x) = 2x3 +ax2 +2bx + 1

substituting the value of -1 in the given p(x),we get

p(x) = 2 * (-1)3 + a * (-1)2 + 2 * b * (-1) + 1

= -2 + a -2b + 1

= -1 + a - 2b

or,a - 2b = 1

also given that 2a - 3b = 4

so we got two equations;

a - 2b = 1 ...(1)

2a - 3b = 4 ...(2)

(1) * 2 = 2a - 4b = 2 ...(3)

(2) * 1 = 2a - 3b = 4 ...(4)

(4) - (3) = [ 2a - 2a ] + [ -3b - (-4b) ] = 4-2

-3b + 4b = 2

therefore b=2

substituting the value of b in (3)

2a - 4b = 2

2a - (4*2) = 2

2a - 8 = 2

2a = 2 + 8

2a = 10

a = 10/2

therefore a = 5

so we get the values of a and b

that is: a = 5 and b=2