Question

If 2 polynomials 2x3 + ax2 + 4x – 12 and x3 + x2 – 2x + a ,leave the same remainder when

divided by ( x – 3 ) , find the value of a and also find the remainder.​

Answer 1

Answer:

value of a equals -3

remainder equals 27

Step-by-step explanation:

By the remainder theorem

If p(x) is divided by ( x - a) then p(a) is the remainder.

Here x - 3 = 0

x = 3.

$now \: for \: the \: first \: polynomial \\ p(3) = 2( {3}^{3}) + a( {3})^{2} + 4(3) - 12 \\ = 2(27) + 9a + 12 - 12 \\ 54 + 9a.$

now for the second polynomial

$p(3) = {3}^{3} + {3}^{2} - 2(3) + a \\ = 27 + 9 - 6 + a \\ 30 + a$

given when two polynomial is divided by (x-3) leaves a same remainder. so

$54 + 9a = 30 + a \\ 54 - 30 = a - 9a \\24 = - 8a \\ a = \frac{24}{ - 8} = - 3 \\ a \: = - 3$

$remainder \: is \: \\ 30 + a = 30 - 3 = 27$