Question

find a if the two polynomials x cube + 3 x square - 9 and 2 x cube + 4 X + a leaves the same remainder when divided by X + 3​

Answer 1

Answer:

a = 57

Step-by-step explanation:

Given that;

Two polynomials (x³ + 3x² - 9) and the other is (2x³ + 4x + a) leaves the same remainder when divided by (x+3).

Using remainder theorem, we can get the value of x.

⇒ x + 3 = 0

⇒ x = - 3

Let the remainder be r of both the polynomial when divided by x + 3.

Substituting the value of x = - 3, in the first polynomial, we get ;

⇒ x³ + 3x² - 9 = r

⇒ (-3)³ + 3(-3)² - 9 = r

⇒ - 27 + 27 - 9 = r

⇒ - 9 = r         .... (i)

Substituting the same value if x for second polynomial -

⇒ 2x³ + 4x + a = r

⇒ 2 (-3)³ + 4(-3) + a = r

⇒ 2 * (-27) - 12 + a = r

⇒ -54 - 12 + a = r

⇒ - 66 + a = r.             .... (ii)

Comparing the value of (i) and (ii), we have ;

-66 + a = - 9

⇒ a = - 9 + 66

⇒ a = 57

Hence, the remainder is 57.

Answer 2

Answer:

Required numeric value of a is 57.

Step-by-step explanation:

It is given that the two different polynomials( x^3 + 3x^2 - 9 and 2x^3 + 4x a ) leave the same remainder, if divided by ( x + 3 ).

Let the remainder be r, when x^3 + 3x^2 - 9 is divided by ( x + 3 ).

According to this question, both the polynomials are leaving the same reminder, so the remainder should be r, if 2x^3 + 4x + a is divided by ( x + 3 ), since x^3 + 3x - 9 is leaving r as remainder when divided by ( x + 3 ).

By Remainder theorem : -

( x + 3 ) = { x - ( - 3 ) } . Thus, substituting the value of x as - 3 in the polynomials for the value of r,

Substituting in x^3 + 3x^2 - 9 : -

= > ( - 3 )^3 + 3( - 3 )^2 - 9 = r

= > - 27 + 27 - 9 = r

= > - 9 = r                  ...( i )

Substituting in 2x^3 + 4x + a

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

= > 2( - 27 ) - 12 + a = r

= > - 54 - 12 + a = r

= > - 66 + a = r             ...( ii )

Then, comparing the values of r from ( i ) and ( ii ) : -

= > - 9= r = - 66 + a

= > - 9 = - 66 + a

= > 66 - 9 = a

= > 57 = a

Hence the required numeric value of a is 57.