2. 3.Polynomials
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3.Polynomials
1. Let x be a variable, n be a positive integer and be constants (real
numbers) then is called a
polynomial in variable “x”
2. The exponent of the highest degree term in a polynomial is known as its degree. A
polynomial of degree “0” is called a constant polynomial
i. A polynomial of degree “1” is called linear polynomial
ii. A polynomial of degree “2” is called quadratic polynomial
iii. A polynomial of degree “3” is called cubic polynomial
3. If is a polynomial and is any real number, then the real number obtained by
reducing by in at and is denoted by
4. A real number is a zero of a polynomial if 0
5. A polynomial of degree “n” can have at most “n” real zeros
6. Geometrically the zeros of a polynomial are the co ordinates of the points
where the graph intersects x- axis.
7. The graph of the quadratic Polynomial 0 is a parabola which
opens upwards (U) or downwards ( according as 0 0
i. Here the graph cuts -axis at two distinct points A and A’ In this case the
co-ordinates of A and A’ are the two zeroes of the quadratic polynomial
The parabola can open either upward or downward.
ii. Here, the graph touches axis at exactly one point, i.e., at two coincident
points So the two points A and A’ of case i coincide here to become one
point A. The x- co-ordinate of A is the only zero for the quadratic polynomial
in this case.
3. 3.Polynomials
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iii. Here, the graph is either completely above the axis or completely below
the axis So, it does not cut the axis at any point. So the quadratic
polynomial has no zero in this case.
8. If and are the zeros of a polynomial then
and .
9. If and are the zeros of a cubic polynomial then
, and .
10. The equation of the quadratic polynomial whose roots are and is
11. The equation of the cubic polynomial whose roots are and is
12. Division algorithm: If and are any two polynomials with 0 then
we can find polynomials and such that where
either 0 or degree of degree of if 0
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Multiple choice Questions
1. If and are the zeros of the polynomial 3, then
(a) (b) (c) (d)None
2. If 3 2 , then the degree of
(a)1 (b) (c)7 (d)5
3. If 3 1 then
(a)2 (b) 1 (c) (d)0
4. If and are the zeros of the polynomial 1 then
(a)1 (b) 1 (c)0 (d)None
5. If one zero of the polynomial 13 is reciprocal of the
other then
(a)2 (b) 2 (c)1 (d)-1
6. If the sum of the zeros of the polynomial 2 3 is 6, then the
value of “ is
(a)2 (b)4 (c)-2 (d)-4
7. If and are the zeros of the polynomial then a polynomial
having and as its zeros is
(a) (b) (c) 1 (d) 1
8. The graph of are given below, for some polynomials then the
number of zeros are
(a)2 (b)3 (c)4 (d)5
9. If then zeros of this polynomial
(a)2,3 (b)-2,-3 (c)6,1 (d)-6,-1
10. If and are the zeros of the polynomial and 3 and 2 then
the polynomial
(a) 3 2 (b) 3 2 (c) 3 2 (d)None
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11. The equation of a polynomial whose zeros are 2 and respectively
(a)3 2 (b) 3 2 (c) 2 1 (d)None
12. If a polynomial 3 3 is divided with 2 then the
quotient is
(a) 1 (b) 1 (c)3x-3 (d)None
13. On dividing 3 2 by a polynomial . The quotient and remainder
were 2 and 2 respectively then
(a) 1(b) 2 1 (c) 2 2 1 (d) 1
14. If he zeros of the polynomial 3 1 are then and
(a)1 2 (b)-1,- 2 (c)1 2 (d)None
15. If the polynomial 3 3 is divided by another polynomial
1 the remainder comes out to be 1 then and ______
(a)2 (b)-2 (c) 1 (d)1
16. If and are the zeros of the polynomial 1 , then
1 1
(a) 1 (b)1-c (c)c (d)1+c
17. If has no real zeros and 0
(a) 0 (b)c>0 (c)c<0 (d)None
18. If the product of zeros of the polynomial 11 is 4, then
(a) (b (c) (d)
19. If and are the zeros of the polynomial , then
(a) (b (c) (d)
20. If two of the zeros f the cubic polynomial, are each equal to zero,
then the third zero is
(a) (b (c) (d)
21. If two zeros of the polynomial are and , then its third zero
is
(a)1 (b)-1 (c)2 (d)-2
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22. The product of the zeros of is
(a)-4 (b4 (c)6 (d)-6
23. What should be added to the polynomial so that 3 is the zero of the
resulting polynomial
(a)1 (b)2 (c)4 (d)5
24. A quadratic polynomial, the sum of whose zeros is 0 and one zero is 3, is
(a) (b) (c) 3 (d) 3
25. The polynomial which when divided by 1 gves a quotient 2 and
remainder 3 is
(a) 3 3 (b) 3 3
(c) 3 3 (d) 3 3
26. If and are the two zeros of the polynomial is 3 1 then its
third zero is
(a)3 (b)-3 (c)5 (d)-5
27. If one root of the polynomial 13 is reciprocal of the other, then
the value of k is
(a)0 (b)5 (c) (d)6
28. If and are the zeros of the quadratic polynomial then
(a) (b) (c) (d)None
29. If and are the zeros of the polynomial such that
1 then
(a)2 (b)3 (c)4 (d)6
30. If is a zero of 2 then b=
(a)0 (b)6 (c)-6 (d)-3
31. Roots of 2 0 are
(a)Real and rational (b) Real and irrational
(c) Real and equal (d) None
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32. Condition that 1 is a factor of is
(a) (b) 0
(c) (d)
33. If and are the roots of 0 then the equation for which the roots
are is
(a) 0 (b) 0
(c) 0 (d) 0
34. The ratio of the roots of the equation 0 is 1 . Then the value of
is
(a) (b) (c) (d) None
35. The sum of the reciprocals of the roots of the equation 0 ;
0 is
(a) (b) (c) (d)
36. If the product of zeros of 3 is then b=
(a) (b) (c) (d) 1
37. Product of a monomial and a binomial is
(a)Binomial (b)Trinomial (c) Monomial (d) None
38. If the degree of polynomial is 4, then the number of zeros of the polynomial are
(a)2 (b)3 (c) (d) 1
39. The curve never meets axis then the number of solutions
(a)1 (b)2 (c)3 (d) Does not exist
40. If 1 is a factor of 3 then
(a)1 (b)-2 (c) (d) -5
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Answers
1 2 3 4 5 6 7 8 9 10
c d d b a b c d b c
11 12 13 14 15 16 17 18 19 20
a b a c a b c b c c
21 22 23 24 25 26 27 28 29 30
b c b a c b b d d b
31 32 33 34 35 36 37 38 39 40
b d c a a b a c d d
