Vikas Bharati Public School
Class IX Mathematics
Polynomials
1. Which of the following expressions are polynomials? Give reason.
(i)
(ii)
(iii)
2.
Write the coefficient of
(i)
3.
(ii)
in
(iii)
in
Give an example of
(i)
(ii)
(iii)
4.
in
a trinomial of degree 999.
a monomial of degree 17.
a binomial of degree 20.
Verify whether the following are the zeroes of the polynomial, indicated against
them.
(i)
(ii)
(iii)
5 and 3
5.
Evaluate: x4 − x3 + x2 - x + 1 for x = 2
6.
Find the remainder when 2x3 − 5x2 + 9x - 8 is divided by x−3.
7.
Using the remainder theorem, find the remainder when
actual division.
is divided by
and verify the result by
8.
Using factor theorem, show that
9.
For what value of k is the polynomial 2y3 +9y2 + y + k is divisible by x−1?
10.
Show that (x−1), (x−2) and (x−3) are the factors of
x3 − 6x2 + 11x − 6.
11.
Factories: 2a2 + bc − 2ab − ac
12.
Factories: 8ab2 − 18a3
13.
Split the middle term and factories the following:
(i)
(ii)
is a factor of
x2 + 18x + 32
x2 - x − 156
14.
Factories 25x2 + 4y2 + 9z2 + 20xy − 12yz − 30xz. Also write the identity used.
15.
Using suitable identity, evaluate (95)3.
16.
Factories: 32x4 − 500x
17.
Find the product using a suitable identity, write the identity also.
(x − 2y − z) (x2 + 4 y2 + z2 + 2xy + zx − 2 yz )
18.
Factories: (a − 3b)3 + (3b − c)3 + (c−a)3
19.
If x = −2 and y = 1, by using an identity find the value of
(4y2 − 9x2) ( 16y4 + 36x2y2 + 81x4 ). Also write the identity used.
20.
Write the expansion of
21.
Classify the following as linear, quadratic, cubic and biquadratic polynomials:
(i)
(ii)
(iii)
(iv)
22.
6x2 + 7x
5 - z + z3
13y
5m4 + 3m3 − 7m + 9
Write the degree of the following polynomials:
(i)
(ii)
5x3y3 − 3xy2 + 2xy − 5
x3y2 + 4x2y − 3x4 + 2
23.
If x=0 and x=−1are the roots of the polynomial f(x) = 2x3 − 3x2 + ax + b, find
the value of a and b.
24.
The polynomials (ax3 + 3x2 − 3) and (2x3 − 5x + a)when divided by (x − 4) = 1
leave the same remainder. Find the value of a.
25.
If (x3 + ax2 + bx + 6) has (x − 2) as a factor and leaves a remainder 3 when
divided by (x − 3), find the values of a and b.
26.
Find the integral roots of the polynomial f(x) = x3 + 6x2 + 11x + 6.
27.
Without actual division show that f(x) = (x3 − 3x2 − 13x + 15) is exactly
divisible by g(x) = (x2 + 2x − 3).
28.
Find the values of a and b so that the polynomial (x3 − 10x2 + ax + b) is exactly
divisible by (x − 1) and (x − 2).
29.
If (y − 2) and
30.
Using factor theorem, show that (x + y), (y + z) and (z + x) are the factors of
f(x) = (x + y + z)3 − x3 − y3 − z3.
31.
Factories:
32.
are factors of my2 + 5y + n, show that m = n.
(i)
x15 − x10y2 −
x5y4 +
y6
(ii)
25a2 − 60ab - 1 + 36b2
Use a suitable identity to factories the following:
(i)
(ii)
x3 + 8y3 + 6x2y + 12xy2
x2 + 2xy + y2 − 1
33.
Factories: x6 − 64
34.
Factories: 216 + 27b3 − 8c3 − 108bc
35.
Use a suitable identity to find the products:
(i)
(ii)
(x + 8) (x − 2)
103 × 96
36.
Factories x3 + 13x2 + 31x − 45. Given that (x + 9) is a factor.
37.
Use suitable identity to prove that:
=
1
38.
Give possible expressions for the length and breadth of the rectangle whose area
is 35y2 + 13y − 12.
39.
If 3x + 2y = 20 and xy =
40.
Without actually calculating the cubes, find the value of 1.53 − 0.93 − 0.63
using a suitable identity.
, find the value of 27x3 + 8y3