Question

Find the remainder when x power 4 +x power 3-2x power 2 + x+ 1 is divided by x-1 by division method & verify using remainder theorem

Answer 1

Answer:

By both method remainder is same .i.e, 2

Step-by-step explanation:

Given Dividend = $x^4+x^3-2x^2+x+1$

Divisor = x - 1

Long division is attached in pics.

remainder we get = 2

According to remainder theorem when a polynomial , p(x) is divided by a liner polynomial of form x - a then p(a) gives the value of remainder,

According to he remainder theorem,

$p(x)=x^4+x^3-2x^2+x+1$

Remainder = p(1) =  $1^4+1^3-2\times1^2+1+1$  = 1 + 1 - 2 + 1 + 1 = 2

Therefore, By both method remainder is same .i.e, 2

Answer 2

Answer:

Dividend = x⁴+x³-2x²+x+1

Divisor = x-1

x-1)x⁴+x³-2x²+x+1(x³+2x²+1

****x⁴-x³

_________________

******2x³-2x²

******2x³-2x²

_________________

************x+1

************x-1

__________________

**Remainder (2)

Quotient = x³+2x+1

Verification by Remainder Theorem:

Let p(x) = x⁴+x³-2x²+x+1

If x⁴+x³-2x²+x+1 divided by (x-1) then the remainder is p(1)

Remainder = p(1)

= 1⁴+1³-2(1)²+1+1

= 1+1-2+1+1

= 2

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