Question

If p(x) = x³ + 3x² - 5x + 8 is divided by the following, then find the remainder with the help of remainder theorem.
(i) x + 1
(ii) 2x - 1
(iii) x + 2
(iv) x - 4
(v) x + 1/3​

Answer 1

Answer:

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

Answer 2

The given question is

p(x) = x³ + 3x² - 5x + 8 is divided by the following, then find the remainder with the help of remainder theorem.

(i) x + 1

(ii) 2x - 1

(iii) x + 2

(iv) x - 4

(v) x + 1/3

(I) let x+1=0

Then x=-1

substitute x=-1 in the equation we get

${( - 1)}^{3} + 3 {( - 1)}^{2} - 5( - 1) + 8 \\ - 1 + 3 + 5 + 8 = 16 - 1 = 15$

(ii) 2x-1=0

2x=1

x=1/2

${ \frac{1}{2} }^{3} + 3( { \frac{1}{2} )}^{2} - 5( \frac{1}{2} ) + 8 \\ \frac{1}{8} + 3( \frac{1}{4} ) - \frac{5}{2} + 8 \\ \frac{1}{8} + \frac{3}{4} - \frac{5}{2} + 8 \\ (\frac{1 + 6 - 20 + 64}{8} ) \\ \frac{71 - 20}{8} \\ \frac{51}{8}$

(iii) x+2=0

x=-2

$({ - 2})^{3} + 3( { - 2})^{2} - 5( - 2) + 8 \\ - 8 + 3(4) + 10 + 8 \\ - 8 + 12 + 10 + 8 = 22$

(iv) x-4=0

x=4

${(4)}^{3} + 3 {(4)}^{2} - 5(4) + 8 \\ 64 + 3(16) - 20 + 8 \\ 64 + 48 - 20 + 8 = 100$

(v) x+1/3=0

x=-1/3

${( \frac{ - 1}{3} )}^{3} + 3 ({ \frac{ - 1}{3} )}^{2} - 5( \frac{ - 1}{3}) + 8 \\ (\frac{ - 1}{27} ) + 3( \frac{1}{9} ) + \frac{5}{3} + 8 \\ \frac{ - 1}{27} + \frac{1}{3} + \frac{5}{3} + 8 \\ ( \frac{ - 1 + 9 + 45 + 72}{27}) \\ \frac{125}{27} \\ \frac{5}{3}$

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