Question

If the polynomial 2x³ + ax² + 3x - 5 and x³ + x² - 4x + a leave the same remainder when divided by x - 2 , find the value of a

Answer 1

Step-by-step explanation:

Let the given polynomials be f(x) and g(x).

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.

Answer 2

${ \large{ \sf{ \underbrace{\underline{\bigstar \:Concept}}}}}$

Let the given polynomials be f(x) and g(x).

When f(x) and g(x) are divided by (x-2) they leave the same remainder.

$\bold{ie}$

(x-2) is a factor of f(x) and g(x).

It means 2 is the zero of f(x) and g(x)

$\bold{So\: that,}$

$\bf{f(2) = g(2)}$

$\sf{2x³+ax²+3x-5}$

$\sf{= x³+x²-4x+a}$

$\sf{2(2)³+a(2)²+3(2)-5}$

$\sf{= 2³+2²-4(2)+a}$

$\sf{2(8)+a(4)+6-5}$

$\sf{= 8+4-8+a}$

$\sf{16+4a+1 = 4+a}$

$\sf{17+4a = 4+a}$

$\sf{4a-a = 4-17}$

$\sf{3a = -13}$

$\underline{\boxed{\sf\purple{a \:=\: -13/3.}}}$

Thankyou :)

Refer the attachment for better understanding ;)