# Question-1

**The graphs of y = p(x) are given in the figure below for some polynomials p(x). Find the number of zeroes of p(x).**

**Solution:**

The graph does not intersects at x -axis.

âˆ´The no. of zeroes of p(x) is 0

# Question-2

**The graphs of y = p(x) are given in the figure below for some polynomials p(x). Find the number of zeroes of p(x).**

**Solution:**

The graph intersects x - axis at one point.

âˆ´ The no. of zeroes of p(x) is One

# Question-3

**The graphs of y = p(x) are given in the figure below for some polynomials p(x). Find the number of zeroes of p(x).**

**Solution:**

The graph intersects x- axis at three points

âˆ´ The no. of zeroes of p(x) is three

# Question-4

**Solution:**

The graph intersects x - axis at two points

âˆ´ The no. of zeroes of p(x) is Two

# Question-5

**Solution:**

The graph intersects x -axis at four points

âˆ´ The no. of zeroes of p(x) is four

# Question-6

**Solution:**

The graph intersects x -axis at three points

âˆ´ The no. of zeroes of p(x) is Three.

# Question-7

**Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.**

(i) x

(iv) 4u

(i) x

^{2}â€“ 2x â€“ 8 (ii) 4s^{2}â€“ 4s + 1 (iii) 6x^{2}â€“ 3 â€“ 7x(iv) 4u

^{2}+ 8u (v) t^{2}â€“ 15 (vi) 3x^{2}â€“ x â€“ 4.**Solution:**

(i) x

^{2}â€“ 2x â€“ 8 = x

^{2}â€“ 4x + 2x â€“ 8

= x(x â€“ 4) + 2(x â€“ 4)

= (x â€“ 4)(x + 2)

Therefore the zeroes of the polynomial x

^{2}â€“ 2x â€“ 8 are {4, -2}.

Relationship between the zeroes and the coefficients of the polynomial:

Sum of the zeroes = â€“ = â€“

Also sum of the zeroes of the polynomial = 4 â€“ 2 = 2.

Product of the zeroes = =

Also product of the zeroes = 4 x â€“2 = â€“ 8

Hence verified.

(ii) 4s

^{2}â€“ 4s + 1 = 4s

^{2}â€“ 2s â€“ 2s + 1

= 2s(2s â€“ 1) â€“ 1(2s â€“ 1)

= (2s â€“ 1)(2s â€“ 1)

= 2(s â€“ 2(s â€“

Therefore the zeroes of the polynomial are

Relationship between the zeroes and the coefficients of the polynomial:

Sum of the zeroes = â€“ = â€“ = 1

Also sum of the zeroes = = 1

Product of the zeroes = =

Also product of the zeroes =

Hence verified.

(iii) 6x

^{2}â€“ 3 â€“ 7x = 6x

^{2}â€“ 7x â€“ 3

= 6x

^{2}â€“ 9x + 2x â€“ 3

= 3x(2x â€“ 3) + 1(2x â€“ 3)

= (2x â€“3)(3x + 1)

= 2(x â€“ 3(x +

= 6(x â€“ (x +

The zeroes of the polynomials are {

Relationship between the zeroes and the coefficients of the polynomial:

Sum of the zeroes = - = â€“ =

Also sum of the zeroes =

Product of the zeroes = =

Also product of the zeroes =

Hence verified.

(iv) 4u

^{2}+ 8u = 4u(u + 2)

= 4[u â€“ 0][u â€“(â€“ 2)]

The zeroes of the polynomials are {0, â€“ 2}

Relationship between the zeroes and the coefficients of the polynomial:

Sum of the zeroes = â€“ = â€“

Also sum of the zeroes =

Product of the zeroes = =

Also product of the zeroes =

Hence verified.

(v) t

^{2}â€“ 15 = (t +

The zeroes of the polynomials are {

Relationship between the zeroes and the coefficients of the polynomial:

Sum of the zeroes = â€“ = â€“

Also sum of the zeroes =

Product of the zeroes = =

Also product of the zeroes =

Hence verified.

(vi) 3x

^{2}â€“ x â€“ 4 = 3x

^{2}â€“ 4x + 3x â€“ 4

= x(3x â€“ 4) + 1(3x â€“ 4)

= (3x â€“ 4)(x + 1)

The zeroes of the polynomials are {

Relationship between the zeroes and the coefficients of the polynomial:

Sum of the zeroes = â€“ = â€“

Also sum of the zeroes =

Product of the zeroes = =

Also product of the zeroes =

Hence verified.

# Question-8

**Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.**

**(i) (ii)****,****(iii) 0, (iv) 1, 1 (v) -, (vi) 4, 1****Solution:**

(i)

Let the quadratic polynomial be ax

^{2}+ bx + c, and its zeroes be Î± and Î²

Given Î± + Î² = =

Î± Î² = -1 =

If a = 4 , b = -1 and c = -4

The quadratic polynomial is 4x

^{2}- x - 4.

**(ii)**

**,**

Let the quadratic polynomial be ax

^{2}+ bx + c, and its zeroes be Î± and Î²

Given Î± + Î² = =

Î± Î² = =

If a = 1, b = - and c =

The quadratic polynomial x

^{2}- x + (or) 3x

^{2}- 3x + 1 .

(iii) 0,

Let the quadratic polynomial be ax

^{2}+ bx + c and the zeroes be Î± + Î²

Given Î± + Î² = 0 =

Î± Î² = =

If a = 1, b = 0 and c =

The quadratic polynomial is x

^{2}+

(iv) 1, 1

Let the quadratic polynomial be ax

^{2}+ bx + c , and its zeroes be Î± + Î²

Given Î± + Î² = = 1

Î± Î² = = 1

âˆ´ If a = 1 , b = -1 and c = 1

âˆ´ The quadratic polynomial is x

^{2}- x + 1.

(v),

Let the quadratic polynomial be ax

^{2}+ bx + c , and its zeroes be Î± + Î²

Given Î± + Î²

**= =**

Î± Î² == .

If a = 4, b = 1 and c = 1

The quadratic polynomial is 4x

^{2}+ x + 1.

(vi) 4, 1

Let the quadratic polynomial be ax

^{2}+ bx + c

Given Î± + Î² = = 4

Î± Î² = = 1

If a = 1, b = -4 and c = 1

âˆ´ The quadratic polynomial is x

^{2}â€“ 4x +1.

# Question-9

**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**^{3}â€“ 3x^{2}+ 5x - 3, g(x) = x^{2}â€“ 2**(ii) p(x) = x**^{4}â€“ 3x^{2}+ 4x + 5, g(x) = x^{2}+ 1 â€“ x**(iii) p(x) = x**^{4}â€“ 5x + 6, g(x) = 2 â€“ x^{2}**Solution:**

(i) p(x) = x

^{3}â€“ 3x

^{2}+ 5x - 3, g(x) = x

^{2}â€“ 2

Quotient is (x - 3)

Remainder is 7x â€“ 9

(ii) p(x) = x

^{4}â€“ 3x

^{2}+ 4x + 5, g(x) = x

^{2}+ 1 â€“ x

Rearrange g(x) as x

^{2}- x + 1

The Quotient is x

^{2}+ x â€“ 3

Remainder is 8

(iii) p(x) = x

^{4}â€“ 5x + 6, g(x) = 2 â€“ x

^{2}

Rearrange g(x) as â€“x

^{2}+ 2

Quotient is â€“x

^{2}â€“ 2

Remainder is â€“5x + 10

# Question-10

**Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial.**

(i) t

(ii) x

(iii) x(i) t

^{2}â€“ 3, 2t^{4}+ 3t^{3}â€“ 2t^{2}â€“ 9t â€“ 12(ii) x

^{2}+ 3x +1, 3x^{4}+ 5x^{3}â€“ 7x^{2}+ 2x + 2(iii) x

^{3}â€“ 3x + 1, x^{5}â€“ 4x^{3}+ x^{2}+ 3x + 1**Solution:**

(i) t

^{2}â€“ 3, 2t

^{4}+ 3t

^{3}â€“ 2t

^{2}â€“ 9t â€“ 12

âˆ´ t

^{2}â€“ 3 is a factor of the polynomial 2t

^{4}+ 3t

^{3}â€“ 2t

^{2}â€“ 9t â€“ 12 .

(ii) x

^{2}+ 3x+1, 3x

^{4}+ 5x

^{3}â€“ 7x

^{2}+ 2x + 2

Hence the polynomial x

^{2}+ 3x+1 is a factor of the second polynomial 3x

^{4}+ 5x

^{3}â€“ 7x

^{2}+ 2x + 2

(iii) x

^{3}â€“ 3x + 1, x

^{5}â€“ 4x

^{3}+ x

^{2}+ 3x + 1

Hence x

^{3}â€“ 3x + 1 is not a factor of x

^{5}â€“ 4x

^{3}+ x

^{2}+ 3x + 1.

# Question-11

**Obtain all other zeroes of 3x**^{4}+ 6x^{3}â€“ 2x^{2}â€“ 10x â€“ 5, if two of its zeroes are and -.**Solution:**

Two zeros are and -.

Since the two zeros are and-

(x -) (x +) = x

^{2}â€“ is a factor of the polynomial (i.e.,) (3x

^{2}â€“ 5)

âˆ´ Divide the given polynomial by 3x

^{2}â€“ 5.

(3x

^{4}+ 6x

^{3}â€“ 2x

^{2}â€“ 10x â€“ 5) = (3x

^{2}â€“ 5) (x

^{2}+ 2x + 1)

= (3x

^{2}â€“ 5) (x + 1)

^{2}

Its zeroes are given by x = -1, x = -1

âˆ´The zeroes of the given polynomial

**3x**are , -, -1, -1.

^{4}+ 6x^{3}â€“ 2x^{2}â€“ 10x â€“ 5# Question-12

**On dividing x**^{3}â€“ 3x^{2}+ x + 2 by a polynomial g(x), the quotient and remainder were x â€“ 2 and â€“2x + 4, respectively. Find g(x).**Solution:**

By the division algorithm,

Dividend = Divisor Ã— Quotient + Remainder.

In this problem f(x)= x

^{3}â€“ 3x

^{2}+ x + 2

**, s**ince (-2x + 4 ) is the remainder, subtract (-2x + 4) from f(x)

then divide the result by(x - 2).

f(x) - (-2x + 4)= x

^{3}â€“ 3x

^{2}+ x + 2 + 2x - 4

=x

^{3}â€“ 3x

^{2}+ 3x - 2

Thus g(x) =x

^{2}â€“ x + 1.

# Question-13

**Give examples of polynomials p(x), g(x), q(x), 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.****Solution:**

If p(x) denotes the dividend, g(x) denotes the divisor, q(x) denotes the quotient, r(x) denotes the remainder.

p(x) = 3x

^{2}â€“ 6x + 12, g(x) = 3, q(x) = x

^{2 }- 2x + 4, r(x) = 0.

p(x) = x

^{3}+ x

^{2}+ x + 1, g(x) = x

^{2}â€“ 1, q(x) = x + 1, r(x) = 2x + 2.

p(x) = 2x

^{3}+ 2x

^{2}â€“ 6x + 2, g(x) = x

^{2}â€“ x â€“ 1, q(x) = 2x + 4, r(x) = 6.