**Notes**
**NCERT Solutions**
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Given below are the Class 10 Maths Worksheet for Polynomials

a) cubic polynomials problems

b) quadartic polynomials Problems

c) Word Problems

** Question 1)** Verify the 1/2 ,1, -2 are zeroes of cubic polynomial 2x

^{3} + x2 -5 + 2. Also verify the relationship between the zeroes and their coefficients.

** Question 2)** p and q are zeroes of the quadratic polynomial x

^{2} – (k + 6)x + 2(2k – 1). Find the value of k if 2(p+q) =pq

Solution
for f(x)=x^{2} – (k + 6)x + 2(2k – 1)

We get,

p+q = k+6

pq = 2(2k-1)

Now

2(p+q) =pq

Therefore,

2(k+6) = 2(2k-1)

or k+6=2k-1

or k=7

** Question 3)** m, n are zeroes of ax

^{2 }– 5x + c. Find the value of a and c if m + n = m. n = 10.

Solution
for f(x)=ax^{2 }– 5x + c

we get

m+n=5/a

mn=c/a

Given m + n = m. n = 10

Therefore,

5/a=10

a=1/2

c/a=10

or c=5

** Question 4)** If a and b are zeroes of the x

^{2} + 7x + 7, find the value of a

^{-1} + b

^{-1 }-2ab

Solution
for f(x)=x^{2} + 7x + 7

we get

a+b=-7

ab=7

Now

a^{-1} + b^{-1 }-2ab

= (a+b-2(ab)^{2})/ab

= -7-98/7=-15

** Question 5)** Find remainder when x

^{3} – ax

^{2} + 6 – a is divided by (x – a).

Solution
Given p(x) =x^{3} – ax^{2} + 6 – a

By remainder theorem

p(a) = 6-a

** Question 6)** If p and q are zeroes of f(x) = x

^{2 }– 5x + k, such that p -q = 1, find the value of k.

Solution
for f(x)=x^{2 }– 5x + k

we get

p+q= 5

pq=k

Now

p -q = 1

(p -q)^{2 }=1

(p+q)^{2 } -4pq=1

25-4k=1

k=6

** Question 7)** Given that two of the zeroes of the cubic polynomial ax

^{3} + bx

^{2} + cx + d are 0, then find the third zero.

Solution
Two zeroes = 0, 0

Let the third zero be k.

The, using relation between zeroes and coefficient of polynomial, we have:

k + 0 + 0 = -b/a

Third zero = k = -b/a

** Question 8)** If the zeroes of the quadratic polynomial x

^{2} + (a + 1) x + b are 2 and -3, then find the value of a and b.

** Question 9)** If one of the zeroes of the cubic polynomial x

^{3} + ax

^{2 }+ bx + c is -1, then find the product of the other two zeroes.

** Question 10)** If a-b, a a+b , are zeroes of x

^{3 }-6x

^{2} + 8x, then find the value of b

** Question 11)** Find a quadratic polynomial, the sum and product of whose zeroes are √2 and -3/2 respectively. Also, find its zeroes.

** Question 12)** Given that the zeroes of the cubic polynomial x

^{3} – 6x

^{2} + 3x + 10 are of the form a, a + b, a + 2b for some real numbers a and b, find the values of a and b as well as the zeroes of the given polynomial.

** Question 13)** Check whether f(x) is a factor of g(x) by dividing f(x) by g(x):

f(x) = 2x

^{4} + 4x

^{3} – 5x

^{2} – 2x + 2, g(x) = x

^{2} + 2x – 2.

** Question 14)** Find the zeroes of the quadratic polynomial x

^{2} + x -12 and verify the relationship between the zeroes and the coefficients.

** Question 15)** If a and b are zeroes of the polynomial f(x) = 2x

^{2} – 7x + 3, find the value of a

^{2 }+ b

^{2}.

** Question 16)** Quadratic polynomial 4x

^{2} + 12x + 9 has zeroes as p and q . Now form a quadratic polynomial whose zeroes are p -1 and q-1

** Question 17)** If one zero of the polynomial 2x

^{2} – 5x – (2k + 1) is twice the other, find both the zeroes of the polynomial and the value of k.

** Question 18)** If m and n are the zeroes of the quadratic polynomial f(x) = x

^{2} – px + q, then find the values of:

a) m

^{2 }+ n

^{2}
b) m

^{-1} + n

^{-1}
Solution
for f(x) = x^{2} – px + q

We get

m+n=p

mn=q

a) m^{2 }+ n^{2} =(m+n)^{2} -2mn= p^{2}-q

b) m^{-1} + n^{-1} = (m+n)/mn = p/q

** Question 19)** If the zeroes of the polynomial f(x) = x

^{3} + 39x + k are in A. P., find the value of k.

** Question 20)** Using division show that 3y

^{2} + 5 is a factor of 6y

^{5 }+ 15y

^{4} + 16y

^{3} + 4y

^{2} + 10y – 35.

Answer
5) (b-a)

6) 6

16) 4x^{2} + 20x + 25

17) -17/9

19) k=-28

Class 10 Maths
Class 10 Science