Question

Find the remainder when x³+3x²+3x+1 is divided by
(i) x+1 (ii) x- 1/2
(iii) x (iv) x+π
(v) 5+2x

Answer 1

Answer:

Step-by-step explanation:

(i) x + 1

Apply remainder theorem

=>x + 1 =0

=> x = - 1

Replace x by – 1 we get

=>x3+3x2 + 3x + 1

=>(-1)3 + 3(-1)2 + 3(-1) + 1

=> -1 + 3 - 3 + 1

=> 0

Remainder is 0

(ii) x –1/2

Apply remainder theorem

=>x – 1/2 =0

=> x = 1/2

Replace x by 1/2 we get

=>x3+3x2 + 3x + 1

=>(1/2)3 + 3(1/2)2 + 3(1/2) + 1

=> 1/8 + 3/4 + 3/2 + 1

Add the fraction taking LCM of denominator we get

=>(1 + 6 + 12 + 8)/8

=>27/8

Remainder is 27/8

(iii) x

Apply remainder theorem

=>x =0

Replace x by 0 we get

=>x3+3x2 + 3x + 1

=>(0)3 + 3(0)2 + 3(0) + 1

=> 0+0 +0 + 1

=> 1

Remainder is 1

(iv) x + π

Apply remainder theorem

=>x + π =0

=> x = - π

Replace x by – π we get

=>x3+3x2 + 3x + 1

=>(- π)3 + 3(-π)2 + 3(-π) + 1

=> - π3 + 3π2 - 3π + 1

Remainder is - π3 + 3π2 - 3π + 1

(v) 5 + 2x

Apply remainder theorem

=>5+2x =0

=> 2x = - 5

=> x = - 5/2

Replace x by – 5/2 we get

=>x3+3x2 + 3x + 1

=>(-5/2)3 + 3(-5/2)2 + 3(-5/2) + 1

=> -125/8 + 75/4 – 15/2 + 1

Add the fraction taking LCM of denominator

=>(-125 + 150 - 60 + 8 )/125

=> -27/8

Remainder is -27/8

Pls mark brainliest if you feel

Answer 2

(i) = 0

(ii) = 27/8

(iii) = 1

(iv) = $-\pi ^3+3\pi ^2-3\pi +1$ 

(v) = -27/8

Step-by-step explanation:

$(i)x+1=0\\x=-1$

put x = -1 in equation (i)

$(-1)^3+3\times(-1)^2+3\times(-1)+1\\-1+3-3+1\\=0$

$(ii)x-\frac{1}{2}=0\\x=\frac{1}{2}$

put x=1/2 in equation (i)

$(\frac{1}{2}) ^3+3\times(\frac{1}{2})^2+3\times(\frac{1}{2})+1\\\frac{1}{8}+3\times\frac{1}{4}+\frac{3}{2}+1\\\frac{1}{8}+\frac{3}{4}+\frac{3}{2}+\frac{1}{1}\\\frac{1+6+12+8}{8}=\frac{27}{8}$

$(iii)x=0$

put x=0 in equation (i)

$0^3+3\times0^2+3\times0+1\\0+0+0+1\\=1$

$(iv)x+\pi =0\\x=-\pi$

put $x=-\pi$ in equation (i)

$(-\pi)^3+3\times(-\pi)^2+3\times(-\pi)+1\\-\pi^3+3\pi^2-3\pi+1$

$(v)5+2x=0\\2x=-5\\x=\frac{-5}{2}$

put $x=\frac{-5}{2}$ in equation-(i)

$(\frac{-5}{2})^33\times(\frac{-5}{2})^2+3(\frac{-5}{2})+1\\=\frac{-27}{8}$