Question

3. The polynomial p(x) = ax3 - 3x2 +4 and
g(x) = 2x3 - 5x + a when divided by (x - 2)
and (x-4) leave the remainders p and q,
respectively. If p-2q = 4, then find the
value of a​

Answer 1

Answer:-

$\red{\bigstar}$The value of a is 38.

• Given:-

ax³ - 3x² + 4 and 2x³ - 5x + a

Divided by (x-2) and (x-4)

Remainder = p and q

• To Find:-

Value of a = ?

• Solution:-

Taking $ax^3 - 3x^2 + 4$

here,

(x - 2)

→ x = 2

Substituting this value of x in the above Equation:-

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

→ $a (8) - 3 (4) + 4$

→ $8a - 12 + 4$

→ $8a -8$

According to the question:-

$\longrightarrow$ 8a - 8 = p

Now,

Taking $2x^3 - 5x + a$

here,

(x - 4)

→ x = 4

Substituting this value of x in the above Equation:-

$2(4)^3 - 5(4) + a$

→ $2 (64) - 20 + a$

→ $128 - 20 + a$

→ $108 + a$

According to the Question:-

$\longrightarrow$ 108 + a = q

• Given that,

p - 2q = 4

Substituting the values:-

$8a - 8 - 2 ( 108 + a) = 4$

$8a - 8 - 216 - 2a = 4$

$6a - 224 = 4$

$6a = 228$

$a = \dfrac{228}{6}$

a = 38

Therefore, the value of a is $\pink{38}$