Question

Find the value of a if the polynomials 2x³ + ax² + 3x - 5 and x³ + x² - 4x - a leave the same remainder when divided by (x - 1)
1. a = -1
2. a = 1
3. a = 2
4. a = -2

Answer 1

Correct Answer - Option 1 : a = -1

Concept:

Remainder Theorem: Let p(x) be any polynomial of degree greater than or equal to one and let a be any real number. If p(x) is divided by the linear polynomial x – a, then the remainder is p(a).

Calculation:

Let,

P(x) =  2x3 + ax2 + 3x - 5

q(x) = x3 + x2 - 4x - a

The remainder of the polynomial, when P(x) is divided by x - 1 will be P(1)

⇒ P(1) = 2(1)³ + a(1)² + 3(1) - 5

⇒ P(1) = a              ......(1)

The remainder of the polynomial, when q(x) is divided by x - 1 will be P(1)

⇒ q(1) = (1)³ + (1)² - 4(1) - a

⇒ q(1) =  1 + 1 - 4 - a

⇒ q(1) = - 2 - a     .......(2)

According to question,

P(1) = q(1)

⇒ a = - 2 - a

⇒ a = -1