Polynomials

Find the zeroes of the polynomial in each of the following: (i) p(x) = x – 4 (ii) g(x) = 3 – 6x (iii) q(x) = 2x –7 (iv) h(y) = 2y

Give an example of a polynomial, which is: (i) monomial of degree 1 (ii) binomial of degree 20 (iii) trinomial of degree 2

Write whether the following statements are True or False. Justify your answer. (i) A binomial can have atmost two terms (ii) Every polynomial is a binomial (iii) A binomial may have degree 5 (iv) Zero of a polynomial is always 0 (v) A polynomial cannot have more than one zero (vi) The degree of the sum of two polynomials each of degree 5 is always 5.

Zero of the zero polynomial is (A) 0 (B) 1 (C) Any real number (D) Not defined

Degree of the zero polynomial is (A) 0 (B) 1 (C) Any natural number (D) Not defined

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

Give possible expressions for the length and breadth of each of thefollowing rectangles, in which their areas are given:

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

(i) 

(ii) 

If x + y + z = 0, show that .

Verify that 

Factorise: 

Factorise each of the following:

(i) 

(ii) 

Verify:

(i) 

(ii) 

Factorise each of the following:

(i)  (ii) 

(iii)  (iv) 

(v) 

Evaluate the following using suitable identities:

(i) (99)3 (ii) (102)3 (iii) (998)3

Write the following cubes in expanded form:

(i)  (ii) 

(iii)  (iv) 

Factorise:

(i) 

(ii) 

Expand each of the following, using suitable identities:

(i)  (ii) 

(iii)  (iv) 

(v)  (vi) 

Factorise the following using appropriate identities:

(i) 9x2 + 6xy + y2

(ii) 

(iii) 

Evaluate the following products without multiplying directly:

(i) 103 x 107 (ii) 95 x 96 (iii) 104 x 96

Use suitable identities to find the following products:

(i)  (ii) 

(iii)  (iv) 

(v) 

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

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

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

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

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

(i) x3 + x2 + x + 1 (ii) x4 + x3 + x2 + x + 1

(iii) x4 + 3x3 + 3x2 + x + 1 (iv) 

Check whether 7 + 3x is a factor of 3x3 + 7x.

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

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

(i) x + 1 (ii)  (iii) x

(iv) x + π (v) 5 + 2x

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

(i) p(x) = x + 5 (ii) p(x) = x - 5 (iii) p(x) = 2x + 5

(iv) p(x) = 3x - 2 (v) p(x) = 3x (vi) p(x) = ax, a ≠ 0

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

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

(i)  (ii) 

(iii) p(x) = x2 - 1, x = 1, - 1 (iv) p(x) = (x + 1) (x - 2), x = - 1, 2

(v) p(x) = x2 , x = 0 (vi) p(x) = lx + m

(vii)  (viii) 

:  

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

(i) p(y) = y2 - y + 1 (ii) p(t) = 2 + t + 2t2 - t3

(iii) p(x) = x3 (iv) p(x) = (x - 1) (x + 1)

:  

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

(i) p(y) = y2 - y + 1 (ii) p(t) = 2 + t + 2t2 - t3

(iii) p(x) = x3 (iv) p(x) = (x - 1) (x + 1)

Find the value of the polynomial  at

(i) x = 0 (ii) x = - 1 (iii) x = 2

Classify the following as linear, quadratic and cubic polynomial:

(i) (ii)  (iii)  (iv)  (v) 

(vi)  (vii) 

Write the degree of each of the following polynomials:

(i)  (ii) 

(iii)  (iv) 3

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

Write the coefficients of in each of the following:

(i)  (ii) 

(iii)  (iv) 

Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.

(i)  (ii)  (iii)

(iv)  (v)