Question

If p(x)= 2x^3+ax^2+3x-5 and q(x)=x^3+x^2-4x+a leaves same remainder when divided byx-2, find the value of a

Answer 1

hello!!
Let the given polynomials be f(x) and g(x).
ATQ, When f(x) and g(x) are divided by (x-2) they leave the same remainder.
I.e (x-2) is a factor of f(x) and g(x). It means 2 is the zero of f(x) and g(x)
So that,
f(2) = g(2)
2x³+ax²+3x-5 = x³+x²-4x+a
2(2)³+a(2)²+3(2)-5 = 2³+2²-4(2)+a

2(8)+a(4)+6-5 = 8+4-8+a

16+4a+1 = 4+a

17+4a = 4+a

4a-a = 4-17

3a = -13

a = -13/3.

Hope it helped you a lot!
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Answer 2

P ( x ) = 2x^3 + ax^2 + 3x - 5 is divided by x - 2

using remainder theorem

P ( 2 ) = remainder

so

p ( 2 ) = 2x^3 + ax^2 + 3x - 5

= 2 ( 2 )^3 + ( 2 )^2 a + 3 ( 2 ) - 5

= 16 + 4a + 6 - 5

= 4a + 17

so remainder = 4a + 17

when q ( x ) = x^3 + x^2 - 4x + a

similarly

q ( 2 ) = remainder

q ( 2 ) = x^3 + x^2 - 4x + a

= ( 2 )^3 + ( 2 )^2 - 4 ( 2 ) + a

= 8 + 4 - 8 + a

= a + 4

so remainder = a + 4

by the condition given in your question we get equation

4a + 17 = a + 4

4a - a = 4 - 17

3a = - 13

a = - 13/3