Let p(x)= x³ + 9x² - x - 105
By factor theorem,x-3 is a factor of p(x),if p(3) = 0
p(3) = 33 + 9(3)³ - 3 - 105
= 27 + 81 - 3 -105
= 108-108
p(3) = 0
Therefore,x-3 is a factor of x³ + 9x² - x - 105

In the given expressin (i), (iii) and (vi) contain only one term.
\(\therefore\) (i), (iii) and (vi) are monomials.
The expression (v) contains two terms. So it is a binomial.
The expression (ii) contains three terms. So it is a trinomial.
The expression (iv) contains four terms. So it is none of the given type. It is a polynomial

The degree of the polynomial is 6

We see that p(x) – q(x) is also a polynomial. Hence the subtraction of any two polynomials is also a polynomial.

Given polynomial is 6ab⁸+5a²b³c²-7ab+4b²c+2 
Degree of each of the terms is given below.
6ab⁸ has degree (1+8) = 9
5a²b³c² has degree (2+3+2) = 7
7ab has degree (1+1) = 2
4b²c has degree (2+1) = 3
The constant term 2 is always regarded as having degree Zero.
The degree of the polynomial  6ab⁸+ 5a²b³c²-7ab + 4b²c+2
= the largest exponent in the polynomial
= 9

Length of the rectangle = 3x + 2 units
Breadth of the rectangle = 3x - 2 units
Area of the rectangle = (3x + 2 ) (3x - 2)
= 9x² - 6x + 6x - 4
= 9x²- 4
When x = 20
Area of the rectangle = 9(20)2 - 4
= 9(400) - 4
= 3600 - 4 = 3596 sq.units.

To multiply (4x – 5) and (2x² + 3x – 6) distribute each term of the first polynomial to every term of the second polynomial. In this case, we need to distribute the terms 4x and –5. Then gather the like terms and combine them:

= 8x³+12x²-24x+10x²-15x+30
= 8x³ +2x²–39x +30
Aliter: You may also use the method of detached coefficients:

∴  (4x –5) (2x²+3x–6) = 8x³ + 2x² – 39x + 30

25m^-2 - 16n² = (5m)² - (4n)²             [ \(\because \) a²-b² = (a - b)(a + b)]
= (5m - 4n) (5m+ 4n)

p(x) = 2x³ - l1x² - 4x + 3
g(x) = 2x + 3
p(x) = 2x³ - 11x² - 4x + 3
\(p\left( \frac { -3 }{ 2 } \right) =2\left( \frac { -3 }{ 2 } \right) ^{ 3 }-11\left( \frac { -3 }{ 2 } \right) ^{ 2 }-4\left( \frac { -3 }{ 2 } \right) +3\)
= \(2\left( \frac { -27 }{ 8 } \right) -11\left( \frac { 9 }{ 4 } \right) +\frac { 12 }{ 2 } +3\)
= \(-\frac { 54 }{ 8 } -\frac { 99 }{ 4 } +6+3\)
= \(-\frac { 27 }{ 4 } -\frac { 99 }{ 4 } +9\)
= \(\frac { 27-99+36 }{ 4 } \)
= \(\frac { 27-99+36 }{ 4 } =\frac { -126+36 }{ 4 } \)
= \(-\frac { 90 }{ 4 } =-\frac { 45 }{ 2 } \)
= \(p\left( \frac { -3 }{ 2 } \right) \)\(\neq \) 0
P(x) is not a multiple of g(x)

5x – 3 = 0 (or) 5x = 3
Then, x = \(\frac{3}{5}\)

Coefficient of x² is \(\pi \)  and Coefficient of x is -1

\({ m }^{ 2 }-\sqrt [ 3 ]{ m } +7m-10\) not a polynomial.
Since the exponent of m is not a whole number \(\left( \sqrt [ 3 ]{ m } ={ m }^{ \frac { 1 }{ 3 } } \right) \) 

\(\frac { 1 }{ { x }^{ 2 } } +3x-4\) is not apolynominal.
Since the exponent of x² is not a whole number but it is \(\left( \frac { 1 }{ 2 } ={ x }^{ -2 } \right) \) negative number