Question

if f (x) =x⁴-2x³+4x²-ax+b is a polynomial such that when it is divided by x-1 and x+1 the remainder are respectively 5 and 19. determine the remainder when f (x) is divided by x-2​

Answer 1

Answer:

Given that the equation

f(x) = x4 – 2x3 + 3x2 – ax +b

When f(x) is divided by (x+1) and (x-1) , the remainders are 19 and 5 respectively .

∴ f(-1) = 19 and f(1) = 5

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

⇒ 1 +2 + 3 + a + b = 19

∴ a + b = 13 ——- (1)

According to given condition f(1) = 5

f(x) = x4 – 2x3 + 3x2 – ax

⇒ 14 – 2 3 + 3 2 – a (1) b = 5

⇒ 1 – 2 + 3 – a + b = 5

∴ b – a = 3 —— (2)

solving equations (1) and (2)

a = 5 and b = 8

Now substituting the values of a and b in f(x) , we get

∴ f(x) = x4 – 2x3 + 3x2 – 5x + 8

Also f(x) is divided by (x-3) so remainder will be f(3)

∴ f(x)= x4 – 2x3 + 3x2 – 5x + 8

⇒ f(3) = 34 – 2 × 33 + 3 × 32 – 5 × 3 + 8

= 81 – 54 + 27 – 15 + 8

= 47

Therefore, f(x) = x4 – 2x3 + 3x2 – ax +b when a=3 and b= 8 is 47

Answer 2

$\tt \purple{When \: f(x)=x^4-2x^3+3x^2-ax+b \: is \: divided \: by \: x+1 \: and \: x-1,} \\ \tt \purple{ \: we \: get \: remainders \: 19 \: and \: 5 \: respectively.} \\ \tt \purple{find \: the \: remainder \: when \: f(x) \: is \: divided \: by \: x-3}$

Solution:-

$\tt \: f(x) = x^4 – 2x^3 + 3x^2 – ax +b$

When f(x) is divided by (x+1) and (x-1) , the remainders are 19 and 5 respectively .

$\tt \: ∴ f(-1) = 19 \: and \: f(1) = 5 \\ \tt \:(-1)4 – 2 (-1)3 + 3(-1)2 – a (-1) + b = 19\\ \tt \:⇒ 1 +2 + 3 + a + b = 19\\ \tt \:∴ a + b = 13 —— (1)$

$\tt \: f(x) = x4 – 2x3 + 3x2 – ax \\ \tt \: ⇒ 14 – 2 3 + 3 2 – a (1) b = 5 \\ \tt \: ⇒ 1 – 2 + 3 – a + b = 5 \\ \tt \: ∴ b – a = 3 —— (2)$

solving (1) and (2)

a = 5 and b = 8

Substituting the values of a and b in f(x)

$\tt \: ∴ f(x) = x^4 – 2x^3 + 3x^2 – 5x + 8$

$\tt \: ∴ f(x)= x^4 – 2x^3 + 3x^2 – 5x + 8 \\ \tt \:⇒ f(3) = 34 – 2 × 33 + 3 × 32 – 5 × 3 + 8\\ \tt \:= 81 – 54 + 27 – 15 + 8\\ \tt \:= 47$

f(x) = x4 – 2x3 + 3x2 – ax +b when a=3 and b= 8 is 47

$\sf \: @WildCat7083$