#### Page No 28:

#### Question 1:

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

(i)

(ii)

(iii)

(iv)

(v)

(v)

#### Answer:

(i) The number of zeroes is 0 as the graph does not cut the *x*-axis at any point.

(ii) The number of zeroes is 1 as the graph intersects the *x*-axis at only 1 point.

(iii) The number of zeroes is 3 as the graph intersects the *x*-axis at 3 points.

(iv) The number of zeroes is 2 as the graph intersects the *x*-axis at 2 points.

(v) The number of zeroes is 4 as the graph intersects the *x*-axis at 4 points.

(vi) The number of zeroes is 3 as the graph intersects the *x*-axis at 3 points.

#### Page No 33:

#### Question 1:

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

#### Answer:

The value of is zero when *x* − 4 = 0 or *x *+ 2 = 0, i.e., when *x* = 4 or *x* = −2

Therefore, the zeroes of are 4 and −2.

Sum of zeroes =

Product of zeroes

The value of 4*s*^{2} − 4*s* + 1 is zero when 2*s* − 1 = 0, i.e.,

Therefore, the zeroes of 4*s*^{2} − 4*s* + 1 areand.

Sum of zeroes =

Product of zeroes

The value of 6*x*^{2} − 3 − 7*x* is zero when 3*x* + 1 = 0 or 2*x *− 3 = 0, i.e., or

Therefore, the zeroes of 6*x*^{2} − 3 − 7*x* are.

Sum of zeroes =

Product of zeroes =

The value of 4*u*^{2} + 8*u* is zero when 4*u* = 0 or *u* + 2 = 0, i.e., *u* = 0 or *u* = −2

Therefore, the zeroes of 4*u*^{2} + 8*u* are 0 and −2.

Sum of zeroes =

Product of zeroes =

The value of *t*^{2} − 15 is zero when or , i.e., when

Therefore, the zeroes of *t*^{2} − 15 are and.

Sum of zeroes =

Product of zeroes =

The value of 3*x*^{2} − *x* − 4 is zero when 3*x* − 4 = 0 or *x* + 1 = 0, i.e., when or *x* = −1

Therefore, the zeroes of 3*x*^{2} − *x* − 4 are and −1.

Sum of zeroes =

Product of zeroes

#### Question 2:

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

#### Answer:

Let the polynomial be , and its zeroes be and .

Therefore, the quadratic polynomial is 4*x*^{2} − *x* − 4.

Let the polynomial be , and its zeroes be and .

Therefore, the quadratic polynomial is 3*x*^{2} − *x* + 1.

Let the polynomial be , and its zeroes be and .

Therefore, the quadratic polynomial is .

Let the polynomial be , and its zeroes be and .

Therefore, the quadratic polynomial is .

Let the polynomial be , and its zeroes be and .

Therefore, the quadratic polynomial is .

Let the polynomial be .

Therefore, the quadratic polynomial is.

#### Page No 36:

#### Question 1:

Divide the polynomial *p*(*x*) by the polynomial *g*(*x*) and find the quotient and remainder in each of the following:

(i)

(ii)

(iii)

#### Answer:

Quotient = *x* − 3

Remainder = 7*x* − 9

Quotient = *x*^{2} + *x* − 3

Remainder = 8

Quotient = −*x*^{2} − 2

Remainder = −5*x* +10

#### Question 2:

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

#### Answer:

**=**

Since the remainder is 0,

Hence, is a factor of .

Since the remainder is 0,

Hence, is a factor of .

Since the remainder ,

Hence, is not a factor of .

#### Question 3:

Obtain all other zeroes of , if two of its zeroes are .

#### Answer:

Since the two zeroes are ,

is a factor of** **.

Therefore, we divide the given polynomial by .

We factorize

Therefore, its zero is given by *x* + 1 = 0

*x* = −1

As it has the term , therefore, there will be 2 zeroes at* x* = −1.

Hence, the zeroes of the given polynomial are, −1 and −1.

#### Question 4:

On dividing by a polynomial *g*(*x*), the quotient and remainder were *x *− 2 and − 2*x* + 4, respectively. Find *g*(*x*).

#### Answer:

*g*(*x*) = ? (Divisor)

Quotient = (*x* − 2)

Remainder = (− 2*x* + 4)

Dividend = Divisor × Quotient + Remainder

*g*(*x*) is the quotient when we divide by

#### Question 4:

On dividing by a polynomial *g*(*x*), the quotient and remainder were *x *− 2 and − 2*x* + 4, respectively. Find *g*(*x*).

#### Answer:

*g*(*x*) = ? (Divisor)

Quotient = (*x* − 2)

Remainder = (− 2*x* + 4)

Dividend = Divisor × Quotient + Remainder

*g*(*x*) is the quotient when we divide by

#### Question 1:

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:

#### Answer:

(i)

Therefore, , 1, and −2 are the zeroes of the given polynomial.

Comparing the given polynomial with , we obtain *a* = 2, *b* = 1, *c* = −5, *d* = 2

Therefore, the relationship between the zeroes and the coefficients is verified.

(ii)

Therefore, 2, 1, 1 are the zeroes of the given polynomial.

Comparing the given polynomial with , we obtain *a* = 1, *b* = −4, *c* = 5, *d* = −2.

Verification of the relationship between zeroes and coefficient of the given polynomial

Multiplication of zeroes taking two at a time = (2)(1) + (1)(1) + (2)(1) =2 + 1 + 2 = 5

Multiplication of zeroes = 2 × 1 × 1 = 2

Hence, the relationship between the zeroes and the coefficients is verified.

#### Question 2:

Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2, − 7, − 14 respectively.

#### Answer:

Let the polynomial be and the zeroes be .

It is given that

If *a* = 1, then *b* = −2, *c* = −7, *d* = 14

Hence, the polynomial is .

#### Question 3:

If the zeroes of polynomial are, find *a* and *b*.

#### Answer:

Zeroes are *a* − *b*, *a* + *a* + *b*

Comparing the given polynomial with , we obtain

*p* = 1, *q* = −3, *r* = 1, *t* = 1

The zeroes are .

Hence, *a* = 1 and *b* = or .

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