Question

When a polynomial p(x) = x⁴ – 2x³ + 3x² – ax + b is divisible by x – 1 and x + 1, the remainders are 5 and 19 respectively. Find the remainder when p(x) is divided by x – 2.

Give♡ = Get Multiple♡​

Answer 1

$\huge \tt{ \underline{ \underline{Question:-}}}$

When a polynomial p(x) = x⁴ – 2x³ + 3x² – ax + b is divisible by x – 1 and x + 1, the remainders are 5 and 19 respectively. Find the remainder when p(x) is divided by x – 2.

$\huge \bf{ \underline { \underline{Solution:- }}}$

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

$\sf \: \therefore \: f(1) = 5 \: and \: f( - 1) = 19$

⇒ (1)⁴ - 2 ×(1)³ + 3 ×(1)² -a×1+b=5

$\bf \: and (-1)⁴ -2 × (-1)³+3×(-1)²- a×(-1) +b=19$

⇒1-2+3-a+b=5

and 1+2+3+a+b

⇒2-a+b=5 and 6+a+b=19

⇒ -a+b =3 and a+b = 13

• Adding these two equations we get

$( - a + b) + (a + b) = 3 + 13$

⇒ 2b=16

⇒ b =8

• Putting b =8 and -a+b=3 we get,

$- a + 8 = 3$

⇒ -a = 3-8

⇒ a = -5

⇒ a=5

• Putting the values of a and b in

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

• The remainder when f(x) is divided by (x-2) is equal to f(2).

So,remainder =f(2)=(2)⁴-2×(2)³+3×(2)²-5 ×2+8}

$\red{ \boxed{=16-16+12-10+8}}$

$\huge \purple \dag \large{ \boxed{=10}}$

Answer 2

Answer:

Question:−‾‾\huge \tt{ \underline{ \underline{Question:-}}}

Question:−

When a polynomial p(x) = x⁴ – 2x³ + 3x² – ax + b is divisible by x – 1 and x + 1, the remainders are 5 and 19 respectively. Find the remainder when p(x) is divided by x – 2.

Solution:−‾‾\huge \bf{ \underline { \underline{Solution:- }}}

Solution:−

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

∴f(1)=5andf(−1)=19\sf \: \therefore \: f(1) = 5 \: and \: f( - 1) = 19∴f(1)=5andf(−1)=19

⇒ (1)⁴ - 2 ×(1)³ + 3 ×(1)² -a×1+b=5

⇒1-2+3-a+b=5

and 1+2+3+a+b

⇒2-a+b=5 and 6+a+b=19

⇒ -a+b =3 and a+b = 13

Adding these two equations we get

(−a+b)+(a+b)=3+13( - a + b) + (a + b) = 3 + 13(−a+b)+(a+b)=3+13

⇒ 2b=16

⇒ b =8

Putting b =8 and -a+b=3 we get,

−a+8=3- a + 8 = 3−a+8=3

⇒ -a = 3-8

⇒ a = -5

⇒ a=5

Putting the values of a and b in

f(x)=x4−2x3+3x2+5x+8\sf \: f(x) = {x}^{4} - 2 {x}^{3} + 3 {x}^{2} + 5x + 8f(x)=x

4

−2x

3

+3x

2

+5x+8

The remainder when f(x) is divided by (x-2) is equal to f(2).

So,remainder =f(2)=(2)⁴-2×(2)³+3×(2)²-5 ×2+8}

=16−16+12−10+8\red{ \boxed{=16-16+12-10+8}}

=16−16+12−10+8

†=10\huge \purple \dag \large{ \boxed{=10}}†

=10

please give me multiple thnx i will back