Question

When a polynomial p(x) =x^4-2x^3+3x^2-ax+b is divisible by x-1andx+1, the remainders are 5&19 respectively . Find the remainder when p(x) is divided by x-2

Answer 1

Given :-

◉ p(x) = x⁴ - 2x³ + 3x² - ax + b, leaves remainder 5 & 19 when divided by x - 1 & x + 1 respectively.

To Find :-

◉ Remainder when p(x) is divided by x - 2

Solution :-

Since, There are two factors and two variables in the polynomial, So at first we need to find the value of a & b.

Given that, x - 1 , x + 1 are factors of p(x)

∴ p(1) = p(-1) = 0

⇒ (1)⁴ - 2(1)³ + 3(1)² - a(1) + b = 0

⇒ 1 - 2 + 3 - a + b = 0

⇒ a - b = 2 ...(1)

Also,

⇒ p(-1) = 0

⇒ (-1)⁴ - 2(-1)³ + 3(-1)² - a(-1) + b = 0

⇒ 1 + 2 + 3 + a + b = 0

⇒ a + b = -6 ...(2)

Adding (1) & (2) , we get

⇒ a - b + a + b = 2 + (-6)

⇒ 2a = -4

⇒ a = -2

Substituting a = -2 in (1), we get

⇒ -2 - b = 2

⇒ - b = 4

⇒ b = -4

Now, p(x) becomes:

➞ x⁴ - 2x³ + 3x² + 2x - 4

Put, x = 2 to find the remainder that would be left when p(x) is divided by x - 2

⇒ (2)⁴ - 2(2)³ + 3(2)² + 2(2) - 4

⇒ 16 - 16 + 12 + 4 - 4

⇒ 12

Hence, The remainder would be 12 when p(x) is divided by x - 2