Chapter 2- Polynomials

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Page No: 28

Exercises 2.1

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.

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

Exercise 2.2

1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
(i) x² – 2x – 8
(ii) 4s² – 4s + 1
(iii) 6x² – 3 – 7x
(iv) 4u² + 8u

 (v) t² – 15

 (vi) 3x² – x – 4

Answer

(i) x² – 2x – 8
= (x - 4) (x + 2)
The value of x² – 2x – 8 is zero when x - 4 = 0 or x + 2 = 0, i.e., when x = 4 or x = -2
Therefore, the zeroes of x² – 2x – 8 are 4 and -2.

Sum of zeroes = 4 + (-2) = 2 = -(-2)/1 = -(Coefficient of x)/Coefficient of x²

Product of zeroes = 4 × (-2) = -8 = -8/1 = Constant term/Coefficient of x²

(ii) 4s² – 4s + 1
= (2s-1)²
The value of 4s² - 4s + 1 is zero when 2s - 1 = 0, i.e., s = 1/2

Therefore, the zeroes of 4s² - 4s + 1 are 1/2 and 1/2.

Sum of zeroes = 1/2 + 1/2 = 1 = -(-4)/4 = -(Coefficient of s)/Coefficient of s²
Product of zeroes = 1/2 × 1/2 = 1/4 = Constant term/Coefficient of s².

(iii) 6x² – 3 – 7x
= 6x² – 7x – 3
= (3x + 1) (2x - 3)
The value of 6x² - 3 - 7x is zero when 3x + 1 = 0 or 2x - 3 = 0, i.e., x = -1/3 or x = 3/2

Therefore, the zeroes of 6x² - 3 - 7x are -1/3 and 3/2.

Sum of zeroes = -1/3 + 3/2 = 7/6 = -(-7)/6 = -(Coefficient of x)/Coefficient of x²
Product of zeroes = -1/3 × 3/2 = -1/2 = -3/6 = Constant term/Coefficient of x².

(iv) 4u² + 8u
= 4u² + 8u + 0
= 4u(u + 2)
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 = 0 + (-2) = -2 = -(8)/4 = -(Coefficient of u)/Coefficient of u²
Product of zeroes = 0 × (-2) = 0 = 0/4 = Constant term/Coefficient of u².

(v) t² – 15
= t² - 0.t - 15
= (t - √15) (t + √15)
The value of t² - 15 is zero when t - √15 = 0 or t + √15 = 0, i.e., when t = √15 or t = -√15
Therefore, the zeroes of t² - 15 are √15 and -√15.Sum of zeroes = √15 + -√15 = 0 = -0/1 = -(Coefficient of t)/Coefficient of t²
Product of zeroes = (√15) (-√15) = -15 = -15/1 = Constant term/Coefficient of u².

(vi) 3x² – x – 4
= (3x - 4) (x + 1)
The value of 3x² – x – 4 is zero when 3x - 4 = 0 and x + 1 = 0,i.e., when x = 4/3 or x = -1
Therefore, the zeroes of 3x² – x – 4 are 4/3 and -1.

Sum of zeroes = 4/3 + (-1) = 1/3 = -(-1)/3 = -(Coefficient of x)/Coefficient of x²

Product of zeroes = 4/3 × (-1) = -4/3 = Constant term/Coefficient of x².

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

(i) 1/4 , -1

(ii) √2 , 1/3 

(iii) 0, √5

(iv) 1,1 

(v) -1/4 ,1/4 

(vi) 4,1

Answer

(i) 1/4 , -1
Let the polynomial be ax² + bx + c, and its zeroes be α and ß
α + ß = 1/4 = -b/a
αß = -1 = -4/4 = c/a
If a = 4, then b = -1, c = -4
Therefore, the quadratic polynomial is 4x² - x -4.

(ii) √2 , 1/3
Let the polynomial be ax² + bx + c, and its zeroes be α and ß
α + ß = √2 = 3√2/3 = -b/a
αß = 1/3 = c/a
If a = 3, then b = -3√2, c = 1
Therefore, the quadratic polynomial is 3x² -3√2x +1.

(iii) 0, √5
Let the polynomial be ax² + bx + c, and its zeroes be α and ß
α + ß = 0 = 0/1 = -b/a
αß = √5 = √5/1 = c/a
If a = 1, then b = 0, c = √5
Therefore, the quadratic polynomial is x² + √5.

(iv) 1, 1
Let the polynomial be ax² + bx + c, and its zeroes be α and ß
α + ß = 1 = 1/1 = -b/a
αß = 1 = 1/1 = c/a
If a = 1, then b = -1, c = 1
Therefore, the quadratic polynomial is x² - x +1.

(v) -1/4 ,1/4
Let the polynomial be ax² + bx + c, and its zeroes be α and ß
α + ß = -1/4 = -b/a
αß = 1/4 = c/a
If a = 4, then b = 1, c = 1
Therefore, the quadratic polynomial is 4x² + x +1.

(vi) 4,1
Let the polynomial be ax² + bx + c, and its zeroes be α and ß
α + ß = 4 = 4/1 = -b/a
αß = 1 = 1/1 = c/a
If a = 1, then b = -4, c = 1
Therefore, the quadratic polynomial is x² - 4x +1.

Page No: 36

Exercise 2.3

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

Answer

(i) p(x) = x³ – 3x² + 5x – 3, g(x) = x² – 2

Quotient = x-3 and remainder 7x - 9

(ii) p(x) = x⁴ – 3x² + 4x + 5, g(x) = x² + 1 – x

Quotient = x² + x - 3 and remainder 8

(iii) p(x) = x⁴ – 5x + 6, g(x) = 2 – x²

Quotient = -x² -2 and remainder -5x +10

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

Answer

(i) t² – 3,  2t⁴ + 3t³ – 2t² – 9t – 12

t² – 3 exactly divides  2t⁴ + 3t³ – 2t² – 9t – 12 leaving no remainder. Hence, it is a factor of  2t⁴ + 3t³ – 2t² – 9t – 12.

(ii) x² + 3x + 1, 3x⁴ + 5x³ – 7x² + 2x + 2

x² + 3x + 1 exactly divides 3x⁴ + 5x³ – 7x² + 2x + 2 leaving no remainder. Hence, it is factor of 3x⁴ + 5x³ – 7x² + 2x + 2.

(iii) x³ – 3x + 1, x⁵ – 4x³ + x² + 3x + 1

x³ – 3x + 1 didn't divides exactly x⁵ – 4x³ + x² + 3x + 1 and leaves 2 as remainder. Hence, it not a factor of x⁵ – 4x³ + x² + 3x + 1.

3. Obtain all other zeroes of 3x⁴ + 6x³ – 2x² – 10x – 5, if two of its zeroes are √(5/3)

and - √(5/3).

Answer

p(x) = 3x⁴ + 6x³ – 2x² – 10x – 5
Since the two zeroes are √(5/3) and - √(5/3).

We factorize x² + 2x + 1

= (x + 1)²

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

x = -1

As it has the term (x + 1)² , therefore, there will be 2 zeroes at x = - 1.
Hence, the zeroes of the given polynomial are √(5/3) and - √(5/3), - 1 and - 1.

4.  On dividing x³ - 3x² + x + 2 by a polynomial g(x), the quotient and remainder were x - 2 and 
-2x + 4, respectively. Find g(x).

Answer

Here in the given question,

Dividend = x³ - 3x² + x + 2

Quotient = x - 2

Remainder = -2x + 4

Divisor = g(x)

We know that,

Dividend = Quotient × Divisor + Remainder

⇒ x³ - 3x² + x + 2 = (x - 2) × g(x) + (-2x + 4)⇒ x³ - 3x² + x + 2 - (-2x + 4) = (x - 2) × g(x)
⇒ x³ - 3x² + 3x - 2 = (x - 2) × g(x)
⇒ g(x) =  (x³ - 3x² + 3x - 2)/(x - 2)

∴ g(x) = (x² - x + 1)

5.Give examples of polynomial p(x), g(x), q(x) and 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

Answer

(i) Let us assume the division of 6x² + 2x + 2 by 2
Here, p(x) = 6x² + 2x + 2
g(x) = 2
q(x) = 3x² + x + 1
r(x) = 0
Degree of p(x) and q(x) is same i.e. 2.
Checking for division algorithm,
p(x) = g(x) × q(x) + r(x)
Or, 6x² + 2x + 2 = 2x (3x² + x + 1)
Hence, division algorithm is satisfied.

(ii) Let us assume the division of x³+ x by x²,
Here, p(x) = x³ + x
g(x) = x²
q(x) = x and r(x) = x
Clearly, the degree of q(x) and r(x) is the same i.e., 1.
Checking for division algorithm,
p(x) = g(x) × q(x) + r(x)
x³ + x = (x² ) × x + x
x³ + x = x³ + x
Thus, the division algorithm is satisfied.

(iii) Let us assume the division of x³+ 1 by x².
Here, p(x) = x³ + 1
g(x) = x²
q(x) = x and r(x) = 1
Clearly, the degree of r(x) is 0.
Checking for division algorithm,
p(x) = g(x) × q(x) + r(x)
x³ + 1 = (x² ) × x + 1
x³ + 1 = x³ + 1
Thus, the division algorithm is satisfied.

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