        &=
nbsp;           &nbs=
p;  Polynomials

1.Find the zeroes =
of the quadratic polynomial and verify the relationship <=
/pre>between the zeroes=
 and the its coefficient.
            =
   i)        &=
nbsp;    x  + 7 x + 12=
            =
   ii)        &=
nbsp;    6x – 3
            =
   iii)        &=
nbsp;   4  x    + 5x – 2  =
            =
   iv)        &=
nbsp;   abx + ( b –ac) x –bc
 <=
/span>
2) Find a quadrati=
c polynomial each with the given numbers as the sum and <=
/pre>product of its zer=
oes respectively:
            =
   i)        &=
nbsp;    ¼, -1
            =
   ii)        &=
nbsp;    , 1/3
 =
;            &n=
bsp; iii)        &=
nbsp;   0, 
 
3) Divide the poly=
nomial p(x) by the polynomial q(x) and also find the quotient and remai=
nder.
            =
   i)        &=
nbsp;    P(x)=3D6x + 11x– 39 x -65 &n=
bsp;        q(x)=3D  x  + x -1=
            =
   ii)        &=
nbsp;    P(x)=3D 30 x+ 11 x³  =
;-82 x² -12 x + 48 =
      &nbs=
p;     &nb=
sp;q(x)=3D 3x² + 2x -4
            =
   iii)        &=
nbsp;   P(x)=3D 3x ⁴ +5x³ -7x^{2<=
/sup> +2x + 2 , q(x)=3D  x 2}+3x +1
 <=
/span>
4) If two zeroes o=
f the polynomial f(x) =3D x⁴ – 6 x³ – 26 =
x² + 138 x -35 are 
2 +   and 2=
 – , find other zeroes.
 <=
/span>
5) If two zeroes o=
f the polynomial f(x) =3D 2x⁴ + x³ – 14 x²⁼ -19 x -6  are  -2 
and -1 , find othe=
r zeroes.
 <=
/span>
6) If two zeroes o=
f the polynomial f(x) =3D x⁴ -3x³ –  x² +9 x -6  are  -  
and   , find othe=
r zeroes.
 <=
/span>
7) On dividing the=
 polynomial f(x)=3Dx ³ – 3 x² + x+ 2 by a polyn=
omial g(x) , 
the quotient is x-=
2 and the remainder -2x + 4 . Find the polynomial g(x). <=
/span>
8) What must be su=
btracted from 8x ⁴ + 14 x ³– 2 x ²+ =
7x -8 so that the 
resulting polynomi=
al is exactly divisible by 4 x² + 3x – 2.
 <=
/span>
9) What must be ad=
ded to f(x)=3D 4x⁴ + 2x³ – 2x² +x -1=
 so that the resulting 
polynomial is divi=
sible by x² + 2x -3.
 <=
/span>
10)  If two zeroes of the polynomial f(x) =3D 2x=
⁴ -3 x³ – 3 x² +6 x -2  are =
 
-   and =
 , find othe=
r zeroes.
 <=
/span>
 <=
/span>
11) Obtain all zer=
oes of  f(x) =3D  x³ +13 x² +32 x +20<=
span style=3D'mso-spacerun:yes'>  if one of the zeroes 
is -2.<=
/span>
 <=
/span>
12) What must be s=
ubtracted from  x⁴ =
+ 2 x ³– 13 x ²-12x +21  so that the 
resulting polynomi=
al is exactly divisible by  x<=
sup>2 -4x +3?
 <=
/span>
13)  What must be added to f(x) =3D x^{4 + 2x³ –x ² +x -1 so that the resulting <=
/o:p>}
polynomial is divi=
sible by x ² + 2x -3.

14) Write a quadratic polynomial , the sum and product=
 of
whose zeros are 3 and -2. ( CBSE 2008)

1=
5)  Find =
the
zeroes of the quadratic polynomial 5 x² -4-8x and verify the
relationship between the zeroes and the coefficients of the polynomial. ( C=
BSE
2008)

&n=
bsp;
(1 mark questions)

1.
6x + 3x + x +1/2. What is the highest degree?



2. 7x⁴ + 5x³ + 2x² +3x + 5. What is the
highest degree?



3. Find the zeroes of the polynomial x² + 5x + 6



4. Find the zeroes of the polynomial x² + 7x + 12



5. Find the zeroes of the polynomial f(x) =3D 4x² + 5x - 2 



6. Divide the polynomial 3x⁴ - 4x³ - 3x - 1 by x - 1<=
br>


7. If (x-2) is a factor of x² + ax + b and a + b =3D 1, find the
values of a and b



8. Check whether -2 and 2 are zeroes of the polynomial x + 2





 
Answers
1. I) (-4,-3)        =
;     ii)(1/2, - 1/2)   &nbs=
p;            &=
nbsp;  iii (-2/ , 
3/4)   &n=
bsp;        iv ( -b/a, c/b)<=
/o:p>
  2)  =
;  i) k(x ² – ¼ x – 1)        ii<=
span style=3D'mso-spacerun:yes'>  k( x² -  2x + 1/3)  &nb=
sp;       
  iii k( x2 + 5)
3)I)      (quo =3D 6x + 5 re=
m =3D -38 x -60)    =
ii ( quo=3D 10x ² – 3x -12  rem   =3D =
0 )      <=
o:p>
    Iii  ( quo=3D 3x² – 4x + 2 and rem   0  )
4) 7 and -5
5) (  -1/2, 3)
6) ( 1,2)
7) g(x)=3D x ^=
2– x +1
8) 14x -10
9) -61x +65
10)  1 and ½
=
11) -10,-1,-2=
12) 2x-3
13)   X-2
 <=
/span>
 <=
/span>