# Important Questions for CBSE Class 10 Maths Chapter 2 - Polynomials 4 Mark Question

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CBSE Class 10 Maths Chapter-2 Polynomials Important Questions

## 4 Mark Questions

1. Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following.
(i) p(x) = x− 3x+ 5− 3, g(x) = x− 2
(ii) p(x) = x− 3x+ 4+ 5, g(x) = x– + 1
(iii) p(x) = x− 5+ 6, g(x) = 2 − x2

Ans. (i)

Therefore, quotient =x – 3and Remainder =7x – 9
(ii)

Therefore, quotient =x2+x– 3 and, Remainder =8
(iii)

Therefore, quotient =−x2 – 2 and, Remainder =−5x+10

2. Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial.
(i) t2−3, 2t4+3t3−2t2−9t−12
(ii) x2+3x+1, 3x4+5x3−7x2+2x+2
(iii) x3−3x+1, x5−4x3+x2+3x+1

Ans. (i)

Remainder = 0
Hence first polynomial is a factor of second polynomial.
(ii)

Remainder = 0
Hence first polynomial is a factor of second polynomial.
(iii)

Remainder ≠0
Hence first polynomial is not factor of second polynomial.

3. Obtain all other zeroes of (3x4+6x3−2x2−10x−5), if two of its zeroes areand .
Ans. Two zeroes of (3x4+6x3−2x2−10x−5) are and  which means that  is a factor of (3x4+6x3−2x2−10x−5).
Applying Division Algorithm to find more factors we get:

We have p(x) = g(x) × q(x).
⇒ (3x4+6x3−2x2−10x−5) = ()(3x2+6x+3)
= ()3(x2+2x+1) = 3()(x2+x+x+1)
= 3()(x+1)(x+1)
Therefore, other two zeroes of (3x4+6x3−2x2−10x−5) are −1 and −1.

4. On dividing (x3−3x2+x+2) by a polynomial g(x), the quotient and remainder were (x-2) and (-2x+4) respectively. Find g(x).
Ans. Let p(x)=x3−3x2+x+2, q(x) = (x – 2) and r(x) = (–2x+4)
According to Polynomial Division Algorithm, we have
p(x)=g(x).q(x)+r(x)
x3−3x2+x+2 = g(x).(x−2)−2x+4
x3−3x2+x+2+2x−4=g(x).(x−2)
x3−3x2+3x−2=g(x).(x−2)
g(x)=
So, Dividing (x3−3x2+3x−2) by (x−2), we get

Therefore, we have g(x)==x2x+1

5. Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and
(i) deg p(x)=deg q(x)
(ii) deg q(x)=deg r(x)
(iii) deg r(x)=0
Ans. (i) Let p(x)=3x2+3x+6, g(x)=3

So, we can see in this example that deg p(x)=deg q(x)=2
(ii) Let p(x)=x3+5 and g(x)=x2−1

We can see in this example that deg q(x)=deg r(x) =1
(iii) Let p(x)=x2+5x−3, g(x) =x+3

We can see in this example that deg r(x)=0

6. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeros and the coefficients.
(i) x 2 −2x−8
(ii) 4s2−4s+1
(iii) 6x2−3−7x
(iv) 4u2+8u
(v) t 2 −15
(vi) 3x2x−4
Ans. (i) x2−2x−8
Comparing given polynomial with general form ax2+bx+c,
We get a=1, b=-2 and c=-8
We have, x2−2x−8
=x2−4x+2x−8
=x(x−4)+2(x−4)=(x−4)(x+2)
Equating this equal to 0 will find values of 2 zeroes of this polynomial.
(x−4)(x+2)=0
⇒ x=4,−2 are two zeroes.
Sum of zeroes =4 – 2=2 =
Product of zeroes =4×−2=−8 =
(ii) 4s2−4s+1
Here, a=4, b=-4 and c=1
We have, 4s2−4s+1
=4s2−2s−2s+1 = 2s(2s−1)−1(2s−1)
=(2s−1)(2s−1)
Equating this equal to 0 will find values of 2 zeroes of this polynomial.
⇒ (2s−1)(2s−1)=0
s=
Therefore, two zeroes of this polynomial are
Sum of zeroes==1 =  =
Product of Zeroes =
(iii) 6x2−3−7x
Here, a=6, b=-7 and c=-3
We have, 6x2−3−7x
=6x2−7x−3 = 6x2−9x+2x−3
=3x(2x−3)+1(2x−3)=(2x−3)(3x+1)
Equating this equal to 0 will find values of 2 zeroes of this polynomial.
⇒ (2x−3)(3x+1)=0
⇒ x=
Therefore, two zeroes of this polynomial are
Sum of zeroes=
Product of Zeroes =
(iv) 4u2+8u
Here, a=4, b=8 and c=0
4u2+8u = 4u(u+2)
Equating this equal to 0 will find values of 2 zeroes of this polynomial.
⇒ 4u(u+2)=0
⇒ u=0,−2
Therefore, two zeroes of this polynomial are 0,−2
Sum of zeroes =0−2=−2 =
Product of Zeroes =0×−2=0 =
(v) t2−15
Here, a=1, b=0 and c=-15
We have, t2−15
⇒ t2=15
⇒ t=
Therefore, two zeroes of this polynomial are
Sum of zeroes=
Product of Zeroes =
(vi) 3x2x−4
Here, a=3, b=-1 and c=-4
We have, 3x2x−4=3x2−4x+3x−4
=x(3x−4)+1(3x−4)=(3x−4)(x+1)
Equating this equal to 0 will find values of 2 zeroes of this polynomial.
⇒ (3x−4)(x+1)=0
⇒ x=
Therefore, two zeroes of this polynomial are
Sum of zeroes=

Product of Zeroes =

7. Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
(i) , −1
(ii) ,13
(iii) 0,
(iv) 1, 1
(v)
(vi) 4, 1
Ans. (i) , −1
Let α and β are two zeroes of above quadratic polynomial.
α+β = =
α× β=-1  =

Quadratic polynomial which satisfies above conditions= 4x2x−4
(ii)
Let α and β be two zeros of above quadratic polynomial.
α+β=
α× β= which is equal to

Quadratic polynomial which satisfies above conditions= 3x2+1
(iii) 0,
Let α and β be two zeros of above quadratic polynomial.
α+β=0  =
α× β= =

Quadratic polynomial which satisfies above conditions= x2+
(iv) 1, 1
Let α and β be two zeros of above quadratic polynomial.
α+β=1  =
α× β= 1  =

Quadratic polynomial which satisfies above conditions= x2x+1
(v)
Let α and β be two zeros of above quadratic polynomial.
α+β=
α× β= =

Quadratic polynomial which satisfies above conditions= 4x2+x+1
(vi) 4, 1
Let α and β be two zeros of above quadratic polynomial.
α+β=4  =
α× β= 1  =

Quadratic polynomial which satisfies above conditions= x2−4x+1

8. Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case:
(i)
(ii)
Ans. (i) Comparing the given polynomial with , we get
and

= 0

= 0

= 0
and  are the zeroes of
Now,  =
And

=
And  =
(ii) Comparing the given polynomial with , we get
and

= 0

= 0
and  are the zeroes of
Now,  =
And  =
=
And  =

9. Find a cubic polynomial with the sum of the product of its zeroes taken two at a time and the product of its zeroes are  respectively.
Ans. Let the cubic polynomial be  and its zeroes be  and
Then  = 2 =  and

And  =
Here,  and
Hence, cubic polynomial will be

10. If the zeroes of the polynomial  are  find  and
Ans. Since  are the zeroes of the polynomial
=

And

Hence  and .

11. If the two zeroes of the polynomial  are  find other zeroes.
Ans. Since  are two zeroes of the polynomial
Let

Squaring both sides,

Now we divide  by  to obtain other zeroes.

and  are the other factors of
and 7 are other zeroes of the given polynomial.

12. If the polynomial  is divided by another polynomial  the remainder comes out to be  find  and
Ans. Let us divide  by .

Remainder =
On comparing this remainder with given remainder, i.e.

And