Question

Factorise each of the following polynomials using synthetic division: (i) x3 – 3x2 – 10x + 24
(ii) 2x3 – 3x2 – 3x + 2
(iii) -7x + 3 + 4x3
(iv) x3 + x2 – 14x – 24
(v) x3 – 7x + 6
(vi) x3 – 10x2 – x + 10​

Answer 1

Answer:

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Given that f (x) = x4 - 2x3 + 3x2 - ax + b divided by x - 1 and x + 1 leaves remainder 5 and 19.

⇒ f(1) = 5 and f(-1) = 19

So, (1)4 - 2 (1)3 + 3 (1)2 - a(1) + b = 5

⇒ 1 - 2 + 3 - a + b = 5

⇒ - a + b = 5 - 2

⇒ - a + b = 3 ----- (1)

f (-1) = 19

So, (-1)4 - 2 (-1)3 + 3 (-1)2 - a (-1) + b = 19

⇒1 + 2 + 3 + a + b = 19

⇒ a + b = 19 - 6

⇒ a + b = 13 ----- (2)

By adding equation (1) and (2), we get

2b = 16 ⇒ b = 8

⇒ a = 13 - 8 = 5 (by substituting value of b in equation (2))

Hence, f(x) = x4 - 2x3 + 3x2 - 5x + 8

It is also given that f (x) = x4 - 2x3 + 3x2 - ax + b is divided by (x - 3).

Therefore, f(3) = (3)4 - 2 (3)3 + 3 (3)2 - 5 (3) + 8

⇒ f(3) = 81 - 54 + 27 - 15 + 8

⇒ f(3) = 47

Thus, the value of remainder when f (x) = x4 - 2x3 + 3x2 - ax + b is divided by (x - 3) is 47.