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

CBSE Class 10 Maths Important Questions Chapter 2 – Polynomials

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) = x³ − 3x² + 5x − 3, g(x) = x² − 2
(ii) p(x) = x⁴ − 3x² + 4x + 5, g(x) = x² – x + 1
(iii) p(x) = x⁴ − 5x + 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)==x²−x+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) 3x²−x−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) 3x²−x−4
Here, a=3, b=-1 and c=-4
We have, 3x²−x−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 quadratic polynomial be ax²+bx+c
Let α and β are two zeroes of above quadratic polynomial.
α+β = = 
α× β=-1  = 
 
Quadratic polynomial which satisfies above conditions= 4x²−x−4
(ii) 
Let quadratic polynomial be ax²+bx+c
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 quadratic polynomial be ax²+bx+c
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 quadratic polynomial be ax²+bx+c
Let α and β be two zeros of above quadratic polynomial.
α+β=1  = 
α× β= 1  =
 
Quadratic polynomial which satisfies above conditions= x²−x+1
(v) 
Let quadratic polynomial be ax²+bx+c
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 quadratic polynomial be ax²+bx+c
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