Question

If f(x) = x4 –x3 -2x2+x+1 is divided by x-1 then the remainder is ________.

Answer 1

GIVEN :-

• f(x) = x⁴ - x³ - 2x² + x + 1.

TO FIND :-

• The Remainder.

SOLUTION :-

◉ Let x - 1 = 0

➣ x = 1

BY REMAINDER THEOREM ,

➣ f(x) = x⁴ - x³ - 2x² + x + 1.

➣ f(1) = (1)⁴ - (1)³ - 2 × (1)² + 1 + 1

➣ 1 - 1 - 2 × 1 + 1 + 1

➣ 1 - 1 - 2 + 2

➣ 1 + 2 - 2 -1

➣ 3 - 3

➣ 0

Hence the Remainder is 0

ADDITIONAL INFORMATION :-

◉ Remember theorem :- If p(x) is any polynomial of degree greater than or equal to 1 and p(x) is divided by the linear polynomial x - a, Then the Remainder is p(a)

◉ Factor theorem :- x - a is a factor of polynomial p(x) , If p(a) = 0. Also if x ' a is a factor of p(x), Then p(a) = 0.

Answer 2

Answer:

$\huge \bf \red{GIVEN :- }$

f(x) = x⁴ - x³ - 2x² + x + 1.

$\huge \sf \gray{TO \: FIND :- }$

The Remainder.

$\huge \tt \green{SOLUTION :- }$

◉ Let x - 1 = 0

➣ x = 1

BY REMAINDER THEOREM ,

f(x) = x⁴ - x³ - 2x² + x + 1.

f(1) = (1)⁴ - (1)³ - 2 × (1)² + 1 + 1

✵1 - 1 - 2 × 1 + 1 + 1

✵ 1 - 1 - 2 + 2

✵ 1 + 2 - 2 -1

✵ 3 - 3

✵ 0

Hence the Remainder is 0

$\sf \pink{ADDITIONAL \: INFORMATION :-}$

☞ Remember theorem :- If p(x) is any polynomial of degree greater than or equal to 1 and p(x) is divided by the linear polynomial x - a, Then the Remainder is p(a)

☞Factor theorem :- x - a is a factor of polynomial p(x) , If p(a) = 0. Also if x ' a is a factor of p(x), Then p(a) = 0.