Which of the following expressions are polynomials in one variable and which are not?

State reasons for your answer.

(1) 4x² - 3x + 7     

(2) y² + √2       

(3) 3√t + t√2          

(4) y +              

(5) x¹⁰ + y³ + t⁵⁰

Write the coefficients of x² in each of the following:

(1) 2 + x² + x              

(2) 2 - x² + x³             

(3)  × x² + x              

(4) √2x - 1

Give one example each of a binomial of degree 35, and of a monomial of degree 100.

Write the degree of each of the following polynomials:

(1) 5x³ + 4x² + 7x                

(2) 4 - y²               

(3) 5t - √7              

(4) 3

Classify the following as linear, quadratic and cubic polynomials:

(1) x² + x    

(2) x - x³       

(3) y + y² + 4           

(4) 1 + x      

(5) 3t        

(6) r²          

(7) 7x³

Find the value of the polynomial 5x - 4x² + 3 at

(1) x = 0                            

(2) x = -1                        

(3) x = 2

Find p (0), p (1) and p (2) for each of the following polynomials:

(1) p(y) = y² – y + 1                      

(2) p(t) = 2 + t + 2t² – t³

(3) p(x) = x³                                 

(4) p(x) = (x – 1) (x + 1)

Verify whether the following are zeroes of the polynomial, indicated against them.

(1) p(x) = 3x + 1, x = -                       

(2) p(x) = 5x – π, x =               

(3) p(x) = x² – 1, x = 1, –1                                     

(4) p(x) = (x + 1) (x – 2), x = – 1, 2     

(5) p(x) = x², x = 0                        

(6) p(x) = lx + m, x = -

(7) p(x) = 3x² - 1, x = - ,     

(8) p(x) = 2x + 1, x =

Find the zero of the polynomials in each of the following cases:

(1) p(x) = x + 5           

(2) p(x) = x – 5               

(3) p(x) = 2x + 5        

(4) p(x) = 3x – 2

(5) p(x) = 3x              

(6) p(x) = ax, a ≠ 0        

(7) p(x) = cx + d, c ≠ 0, c, d are real numbers.

Find the remainder when x³ + 3x² + 3x + 1 is divided by

(1) x + 1          

(2) x –              

(3) x               

(4) x + π                 

(5) 5 + 2x

Find the remainder when x³ – ax² + 6x – a is divided by x – a.

Check whether 7 + 3x is a factor of 3x² + 7x.

Determine which of the following polynomials has (x + 1) a factor:

(1) x³ + x² + x + 1                                                   

(2) x⁴ + x³ + x² + x + 1

(3) x⁴ + 3x³ + 3x² + x + 1                                    

(4) x³ – x² – (2 + √2) x + √2

Use the Factor Theorem to determine whether g(x) is a factor of p(x) in each of the following cases:

(1) p(x) = 2x³ + x² – 2x – 1, g(x) = x + 1

(2) p(x) = x³ + 3x² + 3x + 1, g(x) = x + 2

(3) p(x) = x³ - 4x² + x + 6, g(x) = x - 3

Find the value of k, if x – 1 is a factor of p(x) in each of the following cases:

(1) p(x) = x² + x + k   

(2) p(x) = 2x² + kx + √2   

(3) p(x) = kx² – √2x + 1    

(4) p(x) = kx² – 3x + k

Find the value of k, if x – 1 is a factor of p(x) in each of the following cases:

(1) p(x) = x² + x + k   

(2) p(x) = 2x² + kx + √2   

(3) p(x) = kx² – √2x + 1    

(4) p(x) = kx² – 3x + k

Factorise:

(1) 12x² – 7x + 1        

(2) 2x² + 7x + 3           

(3) 6x² + 5x – 6           

(4) 3x² – x – 4

 

Factorise:

(1) x³ – 2x² – x + 2    

(2) x³ – 3x² – 9x – 5     

(3) x³ + 13x² + 32x + 20           

(4) 2y³ + y² – 2y – 1

Use suitable identities to find the following products:

(1) (x + 4) (x + 10)                         

(2) (x + 8) (x – 10)                           

(3) (3x + 4) (3x – 5)        

(4) (y² + ) (y² –  )              
(5) (3 – 2x) (3 + 2x)

Evaluate the following products without multiplying directly:

(1) 103 × 107                   

(2) 95 × 96                   

(3) 104 × 96

Factorise the following using appropriate identities:

(1) 9x² + 6xy + y²                   

(2) 4y² – 4y + 1                      

(3) x² –

Expand each of the following, using suitable identities:

(1) (x + 2y + 4z)²                       

(2) (2x – y + z)²                    

(3) (–2x + 3y + 2z)²

(4) (3a – 7b – c)²                    

(5) (–2x + 5y – 3z)²               

(6) [  –  + 1]²

Factorise:

(1) 4x² + 9y² + 16z² + 12xy – 24yz – 16xz

(2) 2x² + y² + 8z² – 2√2 xy + 4√2 yz – 8xz

Write the following cubes in expanded form:

(1) (2x + 1)³               

(2) (2a – 3b)³                

(3) [  + 1]³               

(4) [x – ]³

Evaluate the following using suitable identities:

(1) (99)³                      

(2) (102)³                      

(3) (998)³

Factorise each of the following:

(1) 8a³ + b³ + 12a²b + 6ab²      

(2) 8a³ – b³ – 12a²b + 6ab²      

(3) 27 – 125a³ – 135a + 225a²

(4) 64a³ – 27b³ – 144a² b + 108ab²            

(5) 27p³ –  –  +

Verify:

(1) x³ + y³ = (x + y) (x² – xy + y²)

(2) x³ - y³ = (x - y) (x² + xy + y²)

Factorise each of the following:

(1) 27y³ + 125z³                             

(2) 64m³ – 343n³

[Hint: See Question 9.]

Factorise: 27x³ + y³ + z³ – 9xyz.

Verify that x³ + y³ + z³ – 3xyz = ( ) × (x + y + z) × [(x - y)² + (y - z)² + (z - x)²]

If x + y + z = 0, show that x³ + y³ + z³ = 3xyz.

Without actually calculating the cubes, find the value of each of the following:

(1) (–12)³ + (7)³ + (5)³                        

(2) (28)³ + (–15)³ + (–13)³

Give possible expressions for the length and breadth of each of the following rectangles,

in which their areas are given:     

 

 

 

What are the possible expressions for the dimensions of the cuboids whose volumes are given below?