English IV

Solutions Chapter 2 – Polynomials

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 + 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 4s2 − 4s + 1 is zero when 2s − 1 = 0, i.e.,

Therefore, the zeroes of 4s2 − 4s + 1 areand.

Sum of zeroes = 

Product of zeroes 

The value of 6x2 − 3 − 7x is zero when 3x + 1 = 0 or 2− 3 = 0, i.e., or

Therefore, the zeroes of 6x2 − 3 − 7x are.

Sum of zeroes = 

Product of zeroes = 

The value of 4u2 + 8u is zero when 4u = 0 or u + 2 = 0, i.e., u = 0 or u = −2

Therefore, the zeroes of 4u2 + 8u are 0 and −2.

Sum of zeroes = 

Product of zeroes = 

The value of t2 − 15 is zero when  or , i.e., when 

Therefore, the zeroes of t2 − 15 are  and.

Sum of zeroes =

Product of zeroes = 

The value of 3x2 − x − 4 is zero when 3x − 4 = 0 or x + 1 = 0, i.e., when  or x = −1

Therefore, the zeroes of 3x2 − 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 4x2 − x − 4.

Let the polynomial be , and its zeroes be and .

Therefore, the quadratic polynomial is 3x2 − 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 = 7x − 9

Quotient = x2 + x − 3

Remainder = 8

Quotient = −x2 − 2

Remainder = −5x +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 − 2 and − 2x + 4, respectively. Find g(x).

Answer:

g(x) = ? (Divisor)

Quotient = (x − 2)

Remainder = (− 2x + 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 − 2 and − 2x + 4, respectively. Find g(x).

Answer:

g(x) = ? (Divisor)

Quotient = (x − 2)

Remainder = (− 2x + 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 − ba + 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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