Question

5.
For what value of 'a' of the polynomial x^3+ 2x^2y + axy^2+ 2y^3 divisible by x +y
(A) o
(C) 2.
(B) 1
(D) 3​

Answer 1

Step-by-step explanation:

Given:

${x}^{3} + 2 {x}^{2} y + ax {y}^{2} + 2 {y}^{3}$

and x+y

To find: Find value of 'a'

(A) 0

(B) 2

(C) 1

(D) 3

for which polynomial is divisible by x+y

Solution:

  $x + y) {x}^{3} + 2 {x}^{2} y + ax {y}^{2} + 2 {y}^{3} ( {x}^{2} + xy + (a - 1) {y}^{2} \\ {x}^{3} + {x}^{2} y \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ ( - ) \: \: \: \: \: ( - ) \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ - - - - - - - \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ {x}^{2}y + ax {y}^{2} \\ {x}^{2} y + x {y}^{2} \: \: \\ (- ) \: \: \: \: ( - ) \: \: \: \\ - - - - - - \\ (a - 1)x {y}^{2} + 2 {y}^{3} \: \: \: \: \: \: \: \: \: \\ (a - 1)x {y}^{2} + (a - 1) {y}^{3} \\ ( - ) \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: ( - ) \\ - - - - - - - - - - \\( 3 - a ) {y}^{3}$

The remainder of division is (3-a)y³.

but ATQ the polynomial is completely divisible by x+y

So,

Remainder of division will be zero.

Thus,

$(3 - a) {y}^{3} = 0 \\ \\ 3 - a = 0 \\ \\ a = 3$

Value of a is 3.

Final answer:

If x³+2x²y+axy²+2y³ is completely divisible by x+y than value of a is 3.

Option D is correct.

Hope it helps you.

To learn more on brainly:

1)divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following 1) p(x)=a^3-5a^2+6a-7 g(x)=a^2-2​

https://brainly.in/question/40399222