Let P(x) = kx³-2x²+25x-26
By factor theorem, (x - 1) divides P(x) without remainder, P (1) = 0
P(1) k(1)³ -2 (1)² + 25 (1) - 26 = 0
k-2+25-26 = 0
k-3 = 0
k = 3

Let P(x) = 2x³-5x²-28x+15
By factor theorem, (x-5) is a factor of P(x), if P(5) = 0
P(5) = 2(5)³-5(5)²-28(5)+15
= 2\(\times\)125 - 5\(\times\)25 - 140+15
= 250 - 125 - 140+15
= 265 - 265 = 0
ஃ (x-5) is a factor of 2x³-5x²-28x+15

Let P(x) = x⁴+5x²-5x+1
By factor theorem, (x-1) is a factor of P(x), if P(1) = 0
P(1) = 1⁴+5(1²)-5(1)+1
= 1+5-5+1
= 2≠0
ஃ (x-1) is not a factor of x⁴+5x²-5x+1

Let P(x) = x³+5x²-10x+4
By factor theorem (x-1) is a factor of P(x), if P(1) = 0
P(1) = 1³+5(1²)-10(1)+4
= 1+5-10+4
P(1) = 0
ஃ (x-1) is a factor of x³+5x²-10x+4

p(x) = x³- 3x²+ 4x+ 50
g(x) = x - 3 
p(3) = 33 - 3(3)² + 4(3) + 50
=  27 - 27 + 12 + 50 = 62
\(\therefore\) The Remainder 62

p(x) = 4x³ - 12x² + 14x - 3
g(x) = 2x -1
\(p\left( \frac { 1 }{ 2 } \right) =4\left( \frac { 1 }{ 2 } \right) ^{ 3 }-12\left( \frac { 1 }{ 2 } \right) ^{ 2 }+14\left( \frac { 1 }{ 2 } \right) -3\)
= \(4\times \frac { 1 }{ 8 } -12\times \frac { 1 }{ 4 } +14\times \frac { 1 }{ 2 } -3\)
= \(\frac { 1 }{ 2 } -3+7-3\)
= \(\frac { 1 }{ 2 } -6+7\)
= \(\frac { 1 }{ 2 } +1=\frac { 3 }{ 2 } \)
\(\therefore\) The Remainder \(\frac{3}{2}\)

p(x) = x³ - 2x²- 4x - 1
g(x) = x + 1
p(-1) = (-1)³ - 2(-1)² - 4(-1) - 1
= -1- 2 + 4-1
= 4 - 4 = 0
\(\therefore\) The Remainder = 0

p(x) = x³ - 5x² + 4x - 3
g(x) = x - 2
p(x) = x³ - 5x² + 4x - 3
p(2) = (2)³ - 5(2)²+ 4(2) - 3
= 8 - 5(4)+8 - 3
=  8 - 20 + g - 3
= 16 - 23
= -7
\(\therefore\) P(x) is not a multiple of g(x)

p(x) = 2x³- kx² + 3x + 10
When it is exactly divided by x - 2,
p(2) = 0
2(2)³ - k(2)² + 3(2) + 10 =  0
2(8) -  k(4) + 6 + 10 = 0
16 - k(4) + 6 + 10 = 0
16 -  4k + 6 + 10 =  0
32 - 4k = 0
\(\therefore\) k = \(\frac { 32 }{ 4 } =8\)
The value of k = 8

p(x) = x²⁰¹⁸ + 2018
When it is divided by x - 1,
p( 1) = 1²⁰¹⁸+ 2018
= 1 + 2018
= 2019
The Remainder : 2019

p(x) = 3\(\times\)3 - 4\(\times\)2 + 7x - 5
When it is divided by x + 3
p(-3) = 3(-3)³ - 4(-3)² + 7(-3) - 5
= 3(-27) - 4(9) - 21 - 5
= - 81 - 36 - 21- 5 = -143
\(\therefore\) The Remainder : –143

p(x₁) = 2x³ + ax² + 4x - 12
When it is divided by x - 3,
p(3) = 2(3)³ + a(3)² + 4(3) - 12
= 54 + 9a + 12 - 12
= 54 + 9a .....(R₁)
p(x2) = x³ + x² - 2x + a
When it is divided by x - 3,
p(3) = 3³ + 3² - 2(3) + a
= 27 + 9 -6 +a
= 30 + a  ......(R₂)
The given remainders are same (R₁ = R₂)
\(\therefore\) 54 + 9a = 30 + a
9a - a = 30 - 54
8a = -24
\(\therefore\)  a = - 24/8 = - 3
Consider R₂,
Remainder = 30 - 3
= 27