Question

When f(x)= x⁴-2x³+3x²-ax is divided by x+1 and x-1, we got remainders as 19 and 5 respectively if f(x) is divided by x-3.

[Hint : Ans=47]

→Need full explanation​

Answer 1

Answer:

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 ------- (i)

Again , f(1) = 5

⇒ 14 - 2 × 13 + 3 × 12 - a × 1 b = 5

⇒ 1 - 2 + 3 - a + b = 5

∴ b - a = 3 ------ (ii)

solving eqn (i) and (ii) , we get

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

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

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

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

= 81 - 54 + 27 - 15 + 8 = 47

Step-by-step explanation:

hope this helps......

Answer 2

$\huge{\boxed{\fcolorbox{cyan}{pink} { Correct \:Question}}}$

When f(x)= x⁴- 2x³ + 3x² - ax + b is divided by x+1 and x-1, we got remainders as 19 and 5 respectively if f(x) is divided by x-3.

( u left b value)

$\huge{\boxed{\fcolorbox{cyan}{pink} {Answer}}}$

$\sf{\:The \:given \:Equation}$

$\sf\boxed{x⁴ - 2x³ + 3x² - ax + b}$

$\sf{When \:f(x)= x⁴- 2x³ + 3x²- ax + b \:is \:divided \:by \:x + 1 \:and \:x - 1 , \:we \:got \:remainders \:as \:19 and \:5 \:respectively..............(given)}$

$\sf{ \therefore \:f(-1) = 19 \:and \:f(1) = 5 .......(mentioned)}$

Let's substitute f(-1) = 19 in the given equation !;

$\sf\bold{: \implies x⁴ - 2x³ + 3x² - ax + b = 19}$

$\sf{: \implies (-1)⁴ - 2(-1)³ + 3(-1)² - a(-1) + b = 19}$

$\sf{: \implies 1 - 2(-1) + 3(1) - a(-1) + b = 19}$

$\sf{: \implies 1 + 2 + 3 + a + b = 19}$

$\sf{: \implies 6 + a + b = 19}$

$\sf{: \implies a + b = 19 - 6}$

$\sf{: \implies a + b = 13...............⑴}$

Now ;

Let's substitute f(1) = 5 in the given equation !;

$\sf\bold{: \implies x⁴ - 2x³ + 3x² - ax + b = 5}$

$\sf{: \implies (1)⁴ - 2(1)³ + 3(1)² - a(1) + b = 5}$

$\sf{: \implies 1 - 2(1) + 3(1) - a(1) + b = 5}$

$\sf{: \implies 1 - 2 + 3 - a + b = 5}$

$\sf{: \implies 4 - 2 - a + b = 5}$

$\sf{: \implies 2 - a + b = 5}$

$\sf{: \implies - a + b = 5 - 2}$

$\sf{: \implies - a + b = 3}$

$\sf{: \implies b - a = 3 ...........⑵}$

Let's find a and b value ;

$\sf{: \implies a + b = 13...............⑴}$

$\sf{: \implies b - a = 3 ...........⑵}$

$\sf{: \implies b = 3 + a ...........⑶}$

Now Substituting eq⑶ in eq⑴

$\sf{: \implies a + (3 + a) = 13}$$\sf{\because b = 3 + a}$

$\sf{: \implies 2a + 3 = 13}$

$\sf{: \implies 2a = 13 - 3}$

$\sf{: \implies 2a = 10}$

$\sf{: \implies a = \frac{10}{2}}$

$\sf{: \implies a = \frac{\cancel{10}^{5}}{\cancel{2}_{1}}}$

$\sf{: \implies a = 5}$

Substituting a = 5 in eq ⑵

$\sf{: \implies b - (5) = 3}$

$\sf{: \implies b = 3 + 5}$

$\sf{: \implies b = 8}$

• So the value of a = 5 and b = 8

Now let's substitute a and b Value in EQUATION

$\sf\bold{: \implies x⁴ - 2x³ + 3x² - ax + b = 19}$

$\sf{\:Respectively \:if \:f(x) \:is \:divided \:by \:x - 3}$

• x - 3 = 0
• x = 3 So = f(3)

$\sf{: \implies f(x) = x⁴ - 2x³ + 3x² - ax + b }$

$\sf{: \implies f(x) = (3)⁴ - 2(3)³ + 3(3)² - [5(3)] + 8 }$

$\sf{: \implies f(x) = 81 - 54 + 27 - 15 + 8}$

$\sf{: \implies f(x) = 81 - 54 + 27 - 15 + 8}$

$\sf{: \implies f(x) = 81 - 54 + 27 - 15 + 8}$

$\sf{: \implies f(x) = 116 - 69}$

$\sf{: \implies f(x) = 47}$

$\sf\red{Here \:we \:get \:your \:answer :)}$