# Important Questions for CBSE Class 9 Maths Chapter 2 - Polynomials

## CBSE Class 9 Maths Chapter-2 Important Questions - Free PDF Download

**1 Marks Questions**

**1. Give one example each of a binomial of degree 35, and of a monomial of degree 100.**

**Ans. **The binomial of degree 35 can be.

The binomial of degree 100 can be.

**2. Which of the following expression is a polynomial **

**(a) **

**(b) **

**(c) **

**(d) **

**Ans. (a)**

**3. A polynomial of degree 3 in x has at most **

**(a) 5 terms**

**(b) 3 terms **

**(c) 4 terms**

**(d) 1 term **

**Ans. (b) **3 terms

**4. The coefficient of in the polynomial is **

**(a) 2**

**(b) 3 **

**(c) 1**

**(d) 4**

**Ans. (d) **4

**5. The monomial of degree 50 is **

**(a) **

**(b) **

**(c) x+50 **

**(d) 50**

**Ans. (b) **

**6. Divide f(x) by g(x) and verify the remainder, g(x) = x + 4**

**Ans. **Dividend, divisor = x + 4

Quotient, Remainder = 2

Dividend = Divisor quotient + Remainder

**7. Which of the following expression is a monomial **

**(a) 3 + x**

**(b) **

**(c)**

**(d) None of these **

**Ans. (a) **3 + x

**8. A linear polynomial **

**(a) May have one zero**

**(b) has one and only one zero**

**(c) May have two zero**

**(d) May have more than one zero**

**Ans. (b) **has one and only one zero

**9. If, then the value of P(1) + P(-1) is **

**(a) 0**

**(b) 1 **

**(c) 2**

**(d) – 2**

**Ans. (d) **-2

**10. when polynomial is divided by x + 1, the remainder is **

**(a) 1**

**(b) 0 **

**(c) 8**

**(d) – 6 **

**Ans. (b) **-6

**11. Factories **

**Ans. **

= x (x – 1) – y (x – 1)

= (x – 1) (x – y)

**12. The value of K for which x – 1 is a factor of the polynomial is **

**(a) 0**

**(b) 3 **

**(c) – 3**

**(d) 1 **

**Ans. (c**) – 3

**13. The factors of are**

**(a) (3x – 2) (4x + 3)**

**(b) (12x + 1) (x – 6) **

**(c) (12x – 1) (x + 6)**

**(d) (3x + 2) (4x – 3) **

**Ans. (d) **(3x + 2) (4x – 3)

**14. x ^{3} + y^{3} + z^{3} – 3xyz is **

**(a)**

**(b) **

**(c)**

**(d) **

**Ans. (d) **

**15. The expanded form of is **

**(a) **

**(b) **

**(c) **

**(d) **

**Ans. (c) **

**16. Find the integral zeroes of the polynomial **

**Ans. **Given polynomial** **

For zeros** **

Zeroes of polynomial -1, 1, and -3.

**17. The value of (102) ^{3} is**

**(a) 1061208**

**(b) 1001208**

**(c) 1820058**

**(d) none of these**

**Ans. (a) **1061208

**18. is equal to**

**(a) 3abc**

**(b) 3(a-b) (b-c) (c-a) **

**(c) **

**(d) **

**Ans. (b) **3(a-b) (b-c) (c-a)** **

**19. The zeroes of the polynomial p(x) = x (x-2) (x+3) are **

**(a) 0**

**(b) 0, 2, 3 **

**(c) 0, 2, -3**

**(d) none of these**

**Ans. (c) **0, 2, -3

**20. If (x+1) and (x-1) are factors of Px ^{3}+x^{2}-2x+9 then value of p and q are**

**(a) p = -1, q = 2**

**(b) p = 2, q = -1 **

**(c) p = 2, q = 1**

**(d) p = -2, q = -2 **

**Ans. (b) **p = 2, q = -1

**21. If x+y+z = 0, then **

**(a) xyz**

**(b) 2xyz**

**(c) 3xyz**

**(d) 0**

**Ans. (b)** 2xyz

**22. The value of (x-a) ^{3 }+ (x-b)^{3 }+ (x-c)^{3 }– 3 (x-a) (x-b) (x-c) when a + b + c= 3x, is**

**(a) 3**

**(b) 2**

**(c) 1**

**(d) 0**

**Ans. (c) **1

**23. Factors of x ^{2} + **

**(a) **

**(b) **

**(c) **

**(d) **

**Ans. (b) **

**24. The degree of constant function is **

**(a) 1**

**(b) 2**

**(c) 3**

**(d) 0**

**Ans. (d)** 0

**2 Marks Questions**

**1. Write the coefficients of in each of the following:**

**(i) **

**(ii) **

**(iii) **

**(iv) **

**Ans. (i) **

The coefficient ofin the polynomialis 1.

**(ii) **

The coefficient ofin the polynomialis.

**(iii) **

The coefficient ofin the polynomialis.

**(iv) **

The coefficient ofin the polynomialis 0.

**2. Find the value of the polynomialat**

**(i) **

**(ii) **

**(iii) **

**Ans. (i) **Let.

We need to substitute 0 in the polynomialto get

= 3

Therefore, we conclude that at, the value of the polynomialis 3.

**(ii**) Let.

We need to substitutein the polynomialto get

= -6

Therefore, we conclude that at, the value of the polynomialis

**(iii) **Let.

We need to substitute 0 in the polynomialto get

= -3

Therefore, we conclude that at, the value of the polynomialis.

**3. Find the remainder whenis divided by.**

**Ans. **We need to find the zero of the polynomial.

While applying the remainder theorem, we need to put the zero of the polynomialin the polynomial, to get

= 5a

Therefore, we conclude that on dividing the polynomialby, we will get the remainder as.

**4. Evaluate the following products without multiplying directly:**

**(i) **

**(ii) **

**(iii) **

**Ans. (i) **

We can observe that we can apply the identity

= 11021

Therefore, we conclude that the value of the productis.

**(ii) **

We can observe that we can apply the identity

= 9120

Therefore, we conclude that the value of the productis .

**(iii) **

We can observe that, we can apply the identitywith respect to the expression, to get

= 9984

Therefore, we conclude that the value of the productis.

**5. Factorize the following using appropriate identities:**

**(i) **

**(ii) **

**(iii) **

**Ans. (i) **

We can observe that we can apply the identity

**(ii) **

We can observe that we can apply the identity

**(iii) **

We can observe that we can apply the identity

**6. Verify:**

**(i) **

**(ii)**

**Ans. (i) **

We know that.

Therefore, the desired result has been verified.

**(ii) **

We know that.

Therefore, the desired result has been verified.

**7. Factorize:**

**(i) **

**(ii) **

**Ans.**

**(i) **

We know that.

**.**

**(ii) **

The expression can also be written as

We know that.

**.**

Therefore, we conclude that after factorizing the expression, we get**.**

**8. Factorize: **

**Ans.**

We know that.

Therefore, we conclude that after factorizing the expression, we get**.**

**9. Verify that**

**Ans. **

We know that.

And also, we know that.

Therefore, we can conclude that the desired result is verified

**10. If, show that.**

**Ans. **We know that.

We need to substitutein

, to get

Therefore, the desired result is verified

**11. Without actually calculating the cubes, find the value of each of the following:**

**(i) **

**(ii)**

**Ans. (i) **

Let and

We know that, if then

Here,

**= **

**(ii) **

Let and

We know that, if then

Here,

=

**12. Find the value of K if x – 2 is factor of **

**Ans. **x – 2 is factor of

x – 2 = 0

x = 2

K = -36

**13. Factories the polynomial **

**Ans. **

**14. Without actually Calculating the cubes, find the value of **

**Ans. **

= -1260

**15. If x – 3 and are both factors of px ^{2} + 5x + r, then show that p = r**

**Ans. **

From (1) and (2),

9p+r=p+9r

9p-p=9r-r

8p=8r

P=r

Hence prove.

**16. Show that 5 is a zero of polynomial **

**Ans. **Put x = 5 in

= 250-175-80+5

= 255-255 = 0

is zero of polynomial

**17. Using remainder theorem find the remainder when f(x) is divided by g(x) **

** g(x) = x + 1 **

**Ans. **When f(x) is divided by g(x)

Then remainder f(-1)

= 1+ 1 – 2 = 0

**18. Find K if x + 1 is a factor of **

**Ans. Here **

**is factor of P(x)**

**19. Find the values of m and n if the polynomial and x + 2 as its factors. **

**Ans. **x – 1 and x + 2 are factor of

x = 1, x = -2

m+n-12=0

16+4m+2n-14= 0

4m+2n+2 =0

4m+2n= -2

2m+n= -1------(2)

Subtracting (2) from (1)

-m = 13

Put m = -13 in (1)

-13+n=12

N=12+13=25

**20. Check whether 7+ 3x is a factor of **

**Ans. **Let

7 + 3x is factor of p(x)

Remainder = 0

Remainder =

= 0

Hence 7 + 3x is factor of p(x)

**21. Factories **

**Ans. **

** **

We factories by splitting middle term

**22. Evaluate by using suitable identity **

**Ans. **

her a = 100, b =1

= 10000+200+1

= 10201

**23. Find m and n if x – 1 and x – 2 exactly divide the polynomial **

**Ans. Let **

x – 1 and x – 2 exactly divide p(x)

1+m-n+10=0

m-n+11=0

m-n= -11 -----(1)

8+4m-2n+10=0

4m-2n=-18

2m-n=-9 ----{dividing by 2}

Subtracting eq. (2) form (1). We get

-m=-2

M=2

Put m = 2 in eq. (1). We get

2-n=-11

-n=-11-2

+n=+13

N=13

M =2

**24. Factories **

**Ans. **

= (2a-b) (2a-b) (2b-b)

**25. Evaluate **

**Ans.**

We know that

Take a = 100, b =1

= 1000300-30001

=970299

**26. Find the value of k, if x-1 is factor of P(x) and P(x) = 3x ^{2}+kx+**

**Ans. **x-1 is factor of p(x)

**27. Expand **

**Ans. **

**28. Factories **

**Ans. **

**29. Evaluate **

**Ans. **10595

= (100+5) (100-5)

=10000 -25 = 9975

**30. Using factor theorem check whether g(x) is factor of p(x)**

**p(x) = **

**Ans. **Given g(x) =X-3, X-3=0

Put x=3 in p (x)

= 27+9 -49 =36-36 =0

Remainder =0

By factor theorem g(x) is factor of P (X)

**31. Expand **

**Ans. **

**3 Marks Questions**

**1. Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.**

**(i) **

**(ii) **

**(iii) **

**(iv) **

**(v) **

**Ans. (i) **

We can observe that in the polynomial, we have *x* as the only variable and the powers of *x* in each term are a whole number.

Therefore, we conclude thatis a polynomial in one variable.

**(ii) **

We can observe that in the polynomial, we have *y* as the only variable and the powers of *y* in each term are a whole number.

Therefore, we conclude thatis a polynomial in one variable.

**(iii) **

We can observe that in the polynomial, we have *t* as the only variable and the powers of *t* in each term are not a whole number.

Therefore, we conclude thatis not a polynomial in one variable.

**(iv) **

We can observe that in the polynomial, we have *y* as the only variable and the powers of** ***y* in each term are not a whole number.

Therefore, we conclude thatis not a polynomial in one variable.

**(v) **

We can observe that in the polynomial, we have *x*, *y* and *t* as the variables and the** **powers of *x*, *y* and *t* in each term is a whole number.

Therefore, we conclude that is a polynomial but not a polynomial in one variable.

**2. Write the degree of each of the following polynomials:**

**(i) **

**(ii) **

**(iii) **

**(iv) 3**

**Ans. **

**(i) **

We know that the degree of a polynomial is the highest power of the variable in the polynomial.

We can observe that in the polynomial, the highest power of the variable *x* is 3.

Therefore, we conclude that the degree of the polynomialis 3.

**(ii) **

We know that the degree of a polynomial is the highest power of the variable in the polynomial.

We can observe that in the polynomial, the highest power of the variable *y* is 2.

Therefore, we conclude that the degree of the polynomialis 2.

**(iii) **

We know that the degree of a polynomial is the highest power of the variable in the polynomial.

We observe that in the polynomial, the highest power of the variable *t* is 1.

Therefore, we conclude that the degree of the polynomialis 1.

**(iv) 3**

We know that the degree of a polynomial is the highest power of the variable in the polynomial.

We can observe that in the polynomial 3, the highest power of the assumed variable *x* is 0.

Therefore, we conclude that the degree of the polynomial 3 is 0.

**3. Find, and for each of the following polynomials:**

**(i) **

**(ii) **

**(iii)**

**(iv) **

**Ans. (i) **

At:

At:

At:

**(ii) **

At:

At:

At:

**(iii) **

At:

At:

At:

**(iv) **

At:

At:

At:

**4. Use the Factor Theorem to determine whether g(x) is a factor of p(x) in each of the following cases:**

**(i) **

**(ii) **

**(iii)**

**Ans. **

**(i) **

We know that according to the factor theorem,

We can conclude that *g(x)* is a factor of *p(x)*, if *p(-1)*=0.

= 0

Therefore, we conclude that the *g(x)* is a factor of *p(x)*.

**(ii) **

We know that according to the factor theorem,

We can conclude that *g(x)* is a factor of *p(x)*, if *p(-2)*=0.

= -8+12-6+1

=-1

Therefore, we conclude that the *g(x)* is not a factor of *p(x)*.

**(iii) **

We know that according to the factor theorem,

We can conclude that *g(x)* is a factor of *p(x)*, if *p(3)*=0.

= 27-36+3+6

= 0

Therefore, we conclude that the *g(x)* is a factor of *p(x)*.

**5. Find the value of k, if x – 1 is a factor of p(x) in each of the following cases:**

**(i) **

**(ii) **

**(iii) **

**(iv)**

**Ans. (i) **

We know that according to the factor theorem

.

We conclude that ifis a factor of, then.

K+2=0

K=-2

Therefore, we can conclude that the value of *k* is.

**(ii) **

We know that according to the factor theorem

**.**

We conclude that ifis a factor of, then.

Therefore, we can conclude that the value of *k* is.

**(iii) **

We know that according to the factor theorem

**.**

We conclude that ifis a factor of, then.

Therefore, we can conclude that the value of *k* is.

**(iv) **

We know that according to the factor theorem

**.**

We conclude that ifis a factor of, then.

Therefore, we can conclude that the value of *k* is.

**6. Factorize:**

**(i) **

**(ii) **

**(iii) **

**(iv) **

**Ans. (i) **

Therefore, we conclude that on factorizing the polynomial, we get** .**

**(ii) **

Therefore, we conclude that on factorizing the polynomial, we get** .**

**(iii) **

Therefore, we conclude that on factorizing the polynomial, we get.

**(iv) **

Therefore, we conclude that on factorizing the polynomial, we get.

**7. Use suitable identities to find the following products:**

**(i) **

**(ii) **

**(iii) **

**(iv) **

**(v)**

**Ans. (i) **

We know that.

We need to apply the above identity to find the product

Therefore, we conclude that the productis.

**(ii) **

We know that.

We need to apply the above identity to find the product

Therefore, we conclude that the productis.

**(iii) **

We know that.

We need to apply the above identity to find the product

Therefore, we conclude that the productis.

**(iv) **

We know that.

We need to apply the above identity to find the product

Therefore, we conclude that the productis.

**(v) **

We know that.

We need to apply the above identity to find the product

Therefore, we conclude that the product**is.**

**8. Write the following cubes in expanded form:**

**(i) **

**(ii) **

**(iii) **

**(iv) **

**Ans. **

**(i) **

We know that.

Therefore, the expansion of the expressionis .

**(ii) **

We know that.

Therefore, the expansion of the expressionis .

**(iii) **

We know that.

Therefore, the expansion of the expressionis.

**(iv) **

We know that.

Therefore, the expansion of the expressionis.

**9. Evaluate the following using suitable identities:**

**(i) **

**(ii) **

**(iii)**

**Ans. (i) **** **

Using identity,

= 1000000-1-300(99)

= 999999-29700

= 970299.

**(ii) **

Using identity

= 1000000+8+600(102)

= 1000008+61200

=1061208

**(iii) **

Using identity

= 1000000000-8-6000(998)

= 999999992-5988000

= 994011992

**10. Factorize each of the following:**

**(i) **

**(ii) **

**(iii) **

**(iv) **

**(v) **

**Ans. **

**(i) **

The expression can also be written as

Using identitywith respect to the expression, we get.

Therefore, after factorizing the expression , we get.

**(ii) **

The expression can also be written as

Using identitywith respect to the expression, we get.

Therefore, after factorizing the expression , we get.

**(iii) **

The expression can also be written as

Using identitywith respect to the expression, we get.

Therefore, after factorizing the expression , we get.

**(iv) **

The expression can also be written as

Using identitywith respect to the expression, we get.

Therefore, after factorizing the expression , we get.

**(v) **

Using identitywith respect to the expression, to get.

Therefore, after factorizing the expression, we get.

**11. Give possible expressions for the length and breadth of each of the following rectangles, in which their areas are given:**

**(i) **

**(ii) **

**Ans. **

**(i) **

Therefore, we can conclude that a possible expression for the length and breadth of a rectangle of areais.

**(ii) **

Therefore, we can conclude that a possible expression for the length and breadth of a rectangle of areais.

**12. What are the possible expressions for the dimensions of the cuboids whose volumes are given below?**

**(i) **

**(ii) **

**Ans. **

**(i) **

Therefore, we can conclude that a possible expression for the dimension of a cuboid of volumeis.

**(ii) **

The expression can also be written as

Therefore, we can conclude that a possible expression for the dimension of a cuboid of volumeis.

**13. Using suitable identity expand **

**Ans. **

**14. Using factor theorem factories **

**Ans. **

Put x = 1

Put x=3 f(2)

is factor of f(x)

**15. I thought actual division, prove that the polynomial is exactly divisible by (x – 2)**

**Ans. **Let

Is factor of

Zero of

= 16+16+2-34

=34-34=0

is divisible by x-2

**16. Factories 1 – a ^{2} – b^{2} – 2ab**

**Ans. **

**17. Expand **

**Ans. **

**18. Verify each of the following identities **

**(i) x ^{3} + y^{3} = (x + y) (x^{2} – xy + y^{2}) **

**(ii) x ^{3} – y^{3} = (x – y) (x^{2} + xy + y^{2})**

**Ans. (i) **

Taking R.H.S

Verified

**(ii) **

= L.H.S.

Verified

**19. Using identity (a + b) ^{3} = a^{3} + b^{3} + 3ab (a + b) derive the formula a^{3} + b^{3} = (a + b) (a^{2} – ab + b^{2}) **

**Ans. **given

**20. Factories **

**(i) 64y ^{3} + 125z^{3}**

**(ii) 27m ^{3} – 343n^{3} **

**Ans. **Solution

**(i) **64y^{3} + 125z^{3}

**(ii) **27m^{3} – 343n^{3}

**21. Without actually calculating the cubes. Find the value of (26) ^{3} + (-15)^{3} + (11)^{3} **

**Ans. **Let a = 26, b = -15, c = -11

a + b + c = 26 – 15 – 11 = 0

Then a^{3} + b^{3} + c^{3} = 3abc

=12870

**22. Find the values of m and n so that the polynomial x ^{3}-mx^{2}-13x+n has x-1 and x+3 as factors. **

**Ans. **Let polynomial be** **

1-m-13+n=0

-m+n-12 = 0

-12 =m-n ………(1)

And if x-3 is factor of p(x)

-27-9m+39+n=0

-9m+n+12=0

12=9m+n=0

12=9m-n

Subtracting (1) from (2),

8m=24

m=3

Put m = 3 in (1),

3-n=-12

-n=-12-3

-n=-15

N=15

**23. Prove that x ^{2}+6x+15 has no zero.**

**Ans. **

is positive and 6 is positive

**24. Factories 3 (x+y) ^{2 }- 5(x+y) + 2**

**Ans. **

Let x +y =z

Put z = x+y

**25. The volume of a cuboid is given by the expression 3x ^{3}-12x. Find the possible expressions for its dimensions**

**Ans. **The volume of cuboid is given by** **

Dimensions of the cuboid are given by 3x, (x=2) and (x-2)

x+3 is factor of P(x)

=-27-93+39+n=0

=-9m+n 12=0 (2)

=-9m+n=-12

Subtracting eq. (2) from (1)

8m = 24, m = 3

Put m = 3 in eq(1)

m =3 and n =15

**26. Using remainder theorem factories**

**Ans. **

Coefficient of is 1

Constant =3

31 = 3

We can Put x=3 and () and check

Put= x=1

1 – 3 – 1 + 3 = 0

Remainder =0

**27. If is divisible by y – 2 and leaves remainder 3 when divided by y – 3, find the values of a and b.**

**Ans. **Let

p(y) is divisible by y – 2

Then P (2) = 0

If p (y) is divided by y-3 remainder is 3

9a+3b=-30

3a+b=-10 ---(ii)

Subtracting (i) from (ii)

-a = 3 and a = -3

Put a = -3 in eq (i)

-6+b=-7

B=-7+6

B=-1

**28. Factories x ^{6} – 64**

**Ans. **

**29. The volume of a cuboid is given by the algebraic expression ky ^{2}-6ky+8k. Find the possible expressions for the dimensions of the cuboid.**

**Ans. Given volume of cuboid**

Thus dimension of cuboid

**4 Marks Questions**

**1. Classify the following as linear, quadratic and cubic polynomials:**

**(i) **

**(ii) **

**(iii) **

**(iv) **

**(v) 3 t**

**(vi) **

**(vii) **

**Ans.**

**(i)**

We can observe that the degree of the polynomialis 2.

Therefore, we can conclude that the polynomialis a quadratic polynomial.

**(ii)**

We can observe that the degree of the polynomialis 3.

Therefore, we can conclude that the polynomialis a cubic polynomial.

**(iii)**

We can observe that the degree of the polynomialis 2.

Therefore, the polynomialis a quadratic polynomial.

**(iv)**

We can observe that the degree of the polynomialis 1.

Therefore, we can conclude that the polynomialis a linear polynomial.

**(v)**

We can observe that the degree of the polynomialis 1.

Therefore, we can conclude that the polynomialis a linear polynomial.

**(vi)**

We can observe that the degree of the polynomialis 2.

Therefore, we can conclude that the polynomialis a quadratic polynomial.

**(vii)**

We can observe that the degree of the polynomialis 3.

Therefore, we can conclude that the polynomialis a cubic polynomial.

**2. Verify whether the following are zeroes of the polynomial, indicated against them.**

**(i) **

**(ii) **

**(iii) **

**(iv) **

**(v) **

**(vi) **

**(vii) **

**(viii) **

**Ans. (i)**

We need to check whetheris equal to zero or not.

Therefore, we can conclude thatis a zero of the polynomial.

**(ii)**

We need to check whetheris equal to zero or not.

Therefore, is not a zero of the polynomial .

**(iii)**

We need to check whetheris equal to zero or not.

At

At

Therefore, are the zeros of the polynomial .

**(iv)**

We need to check whetheris equal to zero or not.

At

At

Therefore, are the zeros of the polynomial.

**(v)**

We need to check whetheris equal to zero or not.

Therefore, we can conclude thatis a zero of the polynomial.

**(vi)**

We need to check whetheris equal to zero or not.

Therefore, is a zero of the polynomial.

**(vii)**

We need to check whetheris equal to zero or not.

At

At

Therefore, we can conclude thatis a zero of the polynomialbutis not a zero of the polynomial.

**(viii)**

We need to check whetheris equal to zero or not.

Therefore, is a zero of the polynomial

**3. Find the remainder whenis divided by**

**(i) **

**(ii) **

**(iii) **

**(iv) **

**(v) **

**Ans.**

**(i)**

We need to find the zero of the polynomial.

While applying the remainder theorem, we need to put the zero of the polynomialin the polynomial, to get

=-1+3-3+1

= 0

Therefore, we conclude that on dividing the polynomialby, we will get the remainder as 0.

**(ii)**

We need to find the zero of the polynomial.

While applying the remainder theorem, we need to put the zero of the polynomialin the polynomial, to get

Therefore, we conclude that on dividing the polynomialby, we will get** **the remainder as.

**(iii)**

We need to find the zero of the polynomial.

= 0+0+0+1

= 1

Therefore, we conclude that on dividing the polynomialby, we will get the remainder as 1.

**(iv)**

We need to find the zero of the polynomial.

Therefore, we conclude that on dividing the polynomialby, we will get the remainder as.

**(v)**

We need to find the zero of the polynomial.

**4. Determine which of the following polynomials hasa factor:**

**(i) **

**(ii) **

**(iii) **

**(iv) **

**Ans. (i)**

While applying the factor theorem, we get

=-1+1-1+1

=0

We conclude that on dividing the polynomialby, we get the remainder as0.

Therefore, we conclude thatis a factor of.

**(ii)**

While applying the factor theorem, we get

=1-1+1-1+1

=1

We conclude that on dividing the polynomialby, we will get the remainder as1, which is not 0.

Therefore, we conclude thatis not a factor of.

**(iii)**

While applying the factor theorem, we get

=1-3+3-1+1

=1

We conclude that on dividing the polynomialby, we will get the remainder as 1, which is not 0.

Therefore, we conclude thatis not a factor of.

**(iv)**

While applying the factor theorem, we get

We conclude that on dividing the polynomialby, we will get the remainder as, which is not 0.

Therefore, we conclude thatis not a factor of.

**5. Expand each of the following, using suitable identities:**

**(i) **

**(ii) **

**(iii)**

**(iv)**

**(v)**

**Ans. **

**(i)**

We know that.

We need to apply the above identity to expand the expression.

**(ii)**

We know that.

We need to apply the above identity to expand the expression.

**(iii)**

We know that.

We need to apply the above identity to expand the expression.

**(iv)**

We know that.

We need to apply the above identity to expand the expression.

**(v)**

We know that.

**6. Factorize:**

**(i)**

**(ii)**

**Ans.**

**(i)**

The expression ** **can also be written as

We can observe that, we can apply the identitywith** **respect to the expression, to get

Therefore, we conclude that after factorizing the expression**, **we get.

**(ii)**

We need to factorize the expression.

The expression can also be written as

We can observe that, we can apply the identitywith** **respect to the expression, to get

Therefore, we conclude that after factorizing the expression**, **we get.

**5 Marks Questions**

**1. Find the zero of the polynomial in each of the following cases:**

**(i) **

**(ii) **

**(iii) **

**(iv) **

**(v) **

**(vi) **

**(vii) **

**Ans.**

**(i) **

, we need to find.

On puttingequal to 0, we get

Therefore, we conclude that the zero of the polynomial**is.**

**(ii)**

, we need to find.

On puttingequal to 0, we get

Therefore, we conclude that the zero of the polynomialis5.

**(iii)**

, we need to find.

On puttingequal to 0, we get

Therefore, we conclude that the zero of the polynomial**is.**

**(iv)**

, we need to find.

On puttingequal to 0, we get

Therefore, we conclude that the zero of the polynomial**is.**

**(v)**

, we need to find**.**

On puttingequal to 0, we get

Therefore, we conclude that the zero of the polynomialis0.

**(vi)**

, we need to find.

On puttingequal to 0, we get

Therefore, we conclude that the zero of the polynomialis 0.

**(vii)**

**, **we need to find.

On puttingequal to 0, we get

Therefore, we conclude that the zero of the polynomial

**is.**

**2. Check whetheris a factor of.**

**Ans. **We know that if the polynomialis a factor of, then on dividing the polynomialby, we must get the remainder as 0.

We need to find the zero of the polynomial.

While applying the remainder theorem, we need to put the zero of the polynomial in the polynomial, to get

We conclude that on dividing the polynomialby, we will get the remainder as, which is not 0.

Therefore, we conclude thatis not a factor of**.**

**3. Factorize:**

**(i) **

**(ii) **

**(iii) **

**(iv) **

**Ans.**

**(i)**

We need to consider the factors of 2, which are.

Let us substitute 1 in the polynomial, to get

Thus, according to factor theorem, we can conclude thatis a factor of the polynomial.

Let us divide the polynomialby, to get

Therefore, we can conclude that on factorizing the polynomial, we get .

**(ii)**

We need to consider the factors of, which are.

Let us substitute 1 in the polynomial, to get

Thus, according to factor theorem, we can conclude thatis a factor of the polynomial.

Let us divide the polynomialby, to get

Therefore, we can conclude that on factorizing the polynomial, we get** .**

**(iii)**

We need to consider the factors of 20, which are.

Let us substitutein the polynomial, to get

Thus, according to factor theorem, we can conclude thatis a factor of the polynomial.

Let us divide the polynomialby, to get

Therefore, we can conclude that on factorizing the polynomial, we get.

**(iv)**

We need to consider the factors of, which are.

Let us substitute 1 in the polynomial, to get

Thus, according to factor theorem, we can conclude thatis a factor of the polynomial.

Let us divide the polynomialby, to get

Therefore, we can conclude that on factorizing the polynomial, we get**.**

**4. If x ^{2 }– bx + c = (x + p) (x – q) then factories x^{2} – bxy + cy^{2}**

**Ans. **We have

Equating coefficient of and constant

Substituting these values of b and c in We get

**5. Factories (2x – 3y) ^{3} + (3y – 4z)^{3 }+ (4z – 2x)^{3}**

**Ans. Let **

**6. Factories: **

**Ans. **

=

**7. Factories x ^{6 }+ 8y^{6 }- z^{6}+6x^{2}y^{2}z^{2}**

**Ans. **

**8. Factories: **

**Ans. **Given expression can be written as

Thus,