# Chapter – 2: Polynomials

## Exercise 2.1

**1. The graphs of y = p(x) is given in Fig. 2.10 below, for some polynomials p(x). Find the number of zeroes of p(x), in each case.**

**Solutions:**

**Graphical method to find zeroes:-**

Total number of zeroes in any polynomial equation = total number of times the curve intersects x-axis.

(i) In the given graph, the number of zeroes of p(x) is 0 because the graph is parallel to x-axis does not cut it at any point.

(ii) In the given graph, the number of zeroes of p(x) is 1 because the graph intersects the x-axis at only one point.

(iii) In the given graph, the number of zeroes of p(x) is 3 because the graph intersects the x-axis at any three points.

(iv) In the given graph, the number of zeroes of p(x) is 2 because the graph intersects the x-axis at two points.

(v) In the given graph, the number of zeroes of p(x) is 4 because the graph intersects the x-axis at four points.

(vi) In the given graph, the number of zeroes of p(x) is 3 because the graph intersects the x-axis at three points.

## Exercise 2.2

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

**Solutions:**

**(i) x**^{2}**–2x –8**

**⇒**x^{2}– 4x+2x–8 = x(x–4)+2(x–4) = (x-4)(x+2)

Therefore, zeroes of polynomial equation x^{2}–2x–8 are (4, -2)

Sum of zeroes = 4–2 = 2 = -(-2)/1 = -(Coefficient of x)/(Coefficient of x^{2})

Product of zeroes = 4×(-2) = -8 =-(8)/1 = (Constant term)/(Coefficient of x^{2})

**(ii) 4s**^{2}**–4s+1**

⇒4s^{2}–2s–2s+1 = 2s(2s–1)–1(2s-1) = (2s–1)(2s–1)

Therefore, zeroes of polynomial equation 4s^{2}–4s+1 are (1/2, 1/2)

Sum of zeroes = (½)+(1/2) = 1 = -(-4)/4 = -(Coefficient of s)/(Coefficient of s^{2})

Product of zeros = (1/2)×(1/2) = 1/4 = (Constant term)/(Coefficient of s^{2 })

**(iii) 6x**^{2}**–3–7x**

⇒6x^{2}–7x–3 = 6x^{2 }– 9x + 2x – 3 = 3x(2x – 3) +1(2x – 3) = (3x+1)(2x-3)

Therefore, zeroes of polynomial equation 6x^{2}–3–7x are (-1/3, 3/2)

Sum of zeroes = -(1/3)+(3/2) = (7/6) = -(Coefficient of x)/(Coefficient of x^{2})

Product of zeroes = -(1/3)×(3/2) = -(3/6) = (Constant term) /(Coefficient of x^{2 })

**(iv) 4u**^{2}**+8u**

⇒ 4u(u+2)

Therefore, zeroes of polynomial equation 4u^{2} + 8u are (0, -2).

Sum of zeroes = 0+(-2) = -2 = -(8/4) = = -(Coefficient of u)/(Coefficient of u^{2})

Product of zeroes = 0×-2 = 0 = 0/4 = (Constant term)/(Coefficient of u^{2 })

**(v) t**^{2}**–15**

⇒ t^{2} = 15 or t = ±√15

Therefore, zeroes of polynomial equation t^{2} –15 are (√15, -√15)

Sum of zeroes =√15+(-√15) = 0= -(0/1)= -(Coefficient of t) / (Coefficient of t^{2})

Product of zeroes = √15×(-√15) = -15 = -15/1 = (Constant term) / (Coefficient of t^{2 })

**(vi) 3x**^{2}**–x–4**

⇒ 3x^{2}–4x+3x–4 = x(3x-4)+1(3x-4) = (3x – 4)(x + 1)

Therefore, zeroes of polynomial equation3x^{2} – x – 4 are (4/3, -1)

Sum of zeroes = (4/3)+(-1) = (1/3)= -(-1/3) = -(Coefficient of x) / (Coefficient of x^{2})

Product of zeroes=(4/3)×(-1) = (-4/3) = (Constant term) /(Coefficient of x^{2 })

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

**(i) 1/4 , -1**

**Solution:**

From the formulas of sum and product of zeroes, we know,

Sum of zeroes = α+β

Product of zeroes = α β

Sum of zeroes = α+β = 1/4

Product of zeroes = α β = -1

∴ If α and β are zeroes of any quadratic polynomial, then the quadratic polynomial equation can be written directly as:-

**x**^{2}**–(α+β)x +αβ = 0**

**x**^{2}**–(1/4)x +(-1) = 0**

**4x**^{2}**–x-4 = 0**

**Thus,4x**^{2}**–x–4 is the **quadratic polynomial.

**(ii)**√2, 1/3

**Solution:**

Sum of zeroes = α + β =√2

Product of zeroes = α β = 1/3

∴ If α and β are zeroes of any quadratic polynomial, then the quadratic polynomial equation can be written directly as:-

**x**^{2}**–(α+β)x +αβ = 0**

**x**^{2}** –(**√2**)x + (1/3) = 0**

**3x**^{2}**-3**√2x+1 = 0

**Thus, 3x**^{2}**-3**√2x+1 **is the **quadratic polynomial.

**(iii) 0, √5**

**Solution:**

Given,

Sum of zeroes = α+β = 0

Product of zeroes = α β = √5

∴ If α and β are zeroes of any quadratic polynomial, then the quadratic polynomial equation can be written directly

as:-

**x**^{2}**–(α+β)x +αβ = 0**

**x**^{2}**–(0)x +**√5**= 0**

**Thus, x**^{2}**+**√5 **is the **quadratic polynomial.

**(iv) 1, 1**

**Solution:**

Given,

Sum of zeroes = α+β = 1

Product of zeroes = α β = 1

∴ If α and β are zeroes of any quadratic polynomial, then the quadratic polynomial equation can be written directly as:-

**x**^{2}**–(α+β)x +αβ = 0**

**x**^{2}**–x+1 = 0**

**Thus , x**^{2}**–x+1is the **quadratic polynomial.

**(v) -1/4, 1/4**

**Solution:**

Given,

Sum of zeroes = α+β = -1/4

Product of zeroes = α β = 1/4

**x**^{2}**–(α+β)x +αβ = 0**

**x**^{2}**–(-1/4)x +(1/4) = 0**

**4x**^{2}**+x+1 = 0**

**Thus,4x**^{2}**+x+1 is the **quadratic polynomial.

**(vi) 4, 1**

**Solution:**

Given,

Sum of zeroes = α+β =4

Product of zeroes = αβ = 1

**x**^{2}**–(α+β)x+αβ = 0**

**x**^{2}**–4x+1 = 0**

**Thus, x**^{2}**–4x+1 is the **quadratic polynomial.

## 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:**

**(i)** **p(x) = x**^{3}**-3x**^{2}**+5x–3 , g(x) = x**^{2}**–2**

**Solution:**

Given,

Dividend = p(x) = x^{3}-3x^{2}+5x–3

Divisor = g(x) = x^{2}– 2

Therefore, upon division we get,

Quotient = x–3

Remainder = 7x–9

**(ii) p(x) = x**^{4}**-3x**^{2}**+4x+5 , g(x) = x**^{2}**+1-x**

**Solution:**

Given,

Dividend = p(x) = x^{4 }– 3x^{2 }+ 4x +5

Divisor = g(x) = x^{2} +1-x

Therefore, upon division we get,

Quotient = x^{2 }+ x–3

Remainder = 8

**(iii) p(x) =x**^{4}**–5x+6, g(x) = 2–x**^{2}

**Solution:**

Given,

Dividend = p(x) =x^{4} – 5x + 6 = x^{4 }+0x^{2}–5x+6

Divisor = g(x) = 2–x^{2} = –x^{2}+2

Therefore, upon division we get,

Quotient = -x^{2}-2

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:**

**(i) t**^{2}**-3, 2t**^{4 }**+3t**^{3}**-2t**^{2}**-9t-12**

**Solutions:**

Given,

First polynomial = t^{2}-3

Second polynomial = 2t^{4 }+3t^{3}-2t^{2 }-9t-12

As we can see, the remainder is left as 0. Therefore, we say that, t^{2}-3 is a factor of 2t^{4 }+3t^{3}-2t^{2 }-9t-12.

**(ii)x**^{2}**+3x+1 , 3x**^{4}**+5x**^{3}**-7x**^{2}**+2x+2**

**Solutions:**

Given,

First polynomial = x^{2}+3x+1

Second polynomial = 3x^{4}+5x^{3}-7x^{2}+2x+2

As we can see, the remainder is left as 0. Therefore, we say that, x^{2} + 3x + 1 is a factor of 3x^{4}+5x^{3}-7x^{2}+2x+2.

**(iii) x**^{3}**-3x+1, x**^{5}**-4x**^{3}**+x**^{2}**+3x+1**

**Solutions:**

Given,

First polynomial = x^{3}-3x+1

Second polynomial = x^{5}-4x^{3}+x^{2}+3x+1

As we can see, the remainder is not equal to 0. Therefore, we say that, x^{3}-3x+1 is not a factor of x^{5}-4x^{3}+x^{2}+3x+1 .

**3. Obtain all other zeroes of 3x**^{4}**+6x**^{3}**-2x**^{2}**-10x-5, if two of its zeroes are √(5/3) and – √(5/3).**

**Solutions:**

Since this is a polynomial equation of degree 4, hence there will be total 4 roots.

**√(5/3) and – √(5/3) **are zeroes of polynomial f(x).

**∴**** **(x –**√(5/3)**) (x+**√(5/3) **= x^{2}-(5/3) = 0

**(3x**^{2}**−5)=0,** is a factor of given polynomial f(x).

Now, when we will divide f(x) by (3x^{2}−5) the quotient obtained will also be a factor of f(x) and the remainder will be 0.

Therefore, 3x^{4 }+6x^{3 }−2x^{2 }−10x–5 = (3x^{2 }–5)**(x**^{2}**+2x+1)**

Now, on further factorizing (x^{2}+2x+1) we get,

**x**^{2}**+2x+1** = x^{2}+x+x+1 = 0

x(x+1)+1(x+1) = 0

**(x+1)(x+1) = 0**

So, its zeroes are given by: **x= −1 **and** x = −1.**

Therefore, all four zeroes of given polynomial equation are:

**√(5/3),- √(5/3) , −1 and −1.**

Hence, is the answer.

**4. On dividing x**^{3}**-3x**^{2}**+x+2** **by a polynomial g(x), the quotient and remainder were x–2 and –2x+4, respectively. Find g(x).**

**Solution:**

Given,

Dividend, p(x) = x^{3}-3x^{2}+x+2

Quotient = x-2

Remainder = –2x+4

We have to find the value of Divisor, g(x) =?

As we know,

Dividend = Divisor × Quotient + Remainder

∴ x^{3}-3x^{2}+x+2 = g(x)×(x-2) + (-2x+4)

x^{3}-3x^{2}+x+2-(-2x+4) = g(x)×(x-2)

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

Now, for finding g(x) we will divide x^{3}-3x^{2}+3x-2 with (x-2)

Therefore, **g(x) = (x**^{2}**–x+1)**

**5. Give examples of polynomials 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**

**Solutions:**

According to the division algorithm, dividend p(x) and divisor g(x) are two polynomials, where g(x)≠0. Then we can find the value of quotient q(x) and remainder r(x), with the help of below given formula;

Dividend = Divisor × Quotient + Remainder

∴ p(x) = g(x)×q(x)+r(x)

Where r(x) = 0 or degree of r(x)< degree of g(x).

Now let us proof the three given cases as per division algorithm by taking examples for each.

**(i) deg p(x) = deg q(x)**

Degree of dividend is equal to degree of quotient, only when the divisor is a constant term.

Let us take an example, p(x) = 3x^{2}+3x+3 is a polynomial to be divided by g(x) = 3.

So, (3x^{2}+3x+3)/3 = x^{2}+x+1 = q(x)

Thus, you can see, the degree of quotient q(x) = 2, which also equal to the degree of dividend p(x).

Hence, division algorithm is satisfied here.

**(ii) deg q(x) = deg r(x)**

Let us take an example, p(x) = x^{2 }+ 3 is a polynomial to be divided by g(x) = x – 1.

So, x^{2 }+ 3 = (x – 1)×(x) + (x + 3)

Hence, quotient q(x) = x

Also, remainder r(x) = x + 3

Thus, you can see, the degree of quotient q(x) = 1, which is also equal to the degree of remainder r(x).

Hence, division algorithm is satisfied here.

**(iii) deg r(x) = 0**

The degree of remainder is 0 only when the remainder left after division algorithm is constant.

Let us take an example, p(x) = x^{2 }+ 1 is a polynomial to be divided by g(x) = x.

So, x^{2 }+ 1 = (x)×(x) + 1

Hence, quotient q(x) = x

And, remainder r(x) = 1

Clearly, the degree of remainder here is 0.

Hence, division algorithm is satisfied here.

## Exercise 2.4

**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:**

**(i) 2x**^{3}**+x**^{2}**-5x+2; -1/2, 1, -2**

**Solution:**

Given, p(x) **= **2x^{3}+x^{2}-5x+2

And zeroes for p(x) are = 1/2, 1, -2

∴ p(1/2) = 2(1/2)^{3}+(1/2)^{2}-5(1/2)+2 = (1/4)+(1/4)-(5/2)+2 = 0

p(1) = 2(1)^{3}+(1)^{2}-5(1)+2 = 0

p(-2) = 2(-2)^{3}+(-2)^{2}-5(-2)+2 = 0

Hence, proved 1/2, 1, -2 are the zeroes of 2x^{3}+x^{2}-5x+2.

Now, comparing the given polynomial with general expression, we get;

∴ ax^{3}+bx^{2}+cx+d = 2x^{3}+x^{2}-5x+2

a=2, b=1, c= -5 and d = 2

As we know, if α, β, γ are the zeroes of the cubic polynomial ax^{3}+bx^{2}+cx+d , then;

α +β+γ = –b/a

αβ+βγ+γα = c/a

α βγ = – d/a.

Therefore, putting the values of zeroes of the polynomial,

α+β+γ = ½+1+(-2) = -1/2 = –b/a

αβ+βγ+γα = (1/2×1)+(1 ×-2)+(-2×1/2) = -5/2 = c/a

α β γ = ½×1×(-2) = -2/2 = -d/a

Hence, the relationship between the zeroes and the coefficients are satisfied.

**(ii) x**^{3}**-4x**^{2}**+5x-2** ;**2, 1, 1**

**Solution:**

Given, p(x) = x^{3}-4x^{2}+5x-2

And zeroes for p(x) are 2,1,1.

∴ p(2)= 2^{3}-4(2)^{2}+5(2)-2 = 0

p(1) = 1^{3}-(4×1^{2 })+(5×1)-2 = 0

Hence proved, 2, 1, 1 are the zeroes of x^{3}-4x^{2}+5x-2

Now, comparing the given polynomial with general expression, we get;

∴ ax^{3}+bx^{2}+cx+d = x^{3}-4x^{2}+5x-2

a = 1, b = -4, c = 5 and d = -2

As we know, if α, β, γ are the zeroes of the cubic polynomial ax^{3}+bx^{2}+cx+d , then;

α + β + γ = –b/a

αβ + βγ + γα = c/a

α β γ = – d/a.

Therefore, putting the values of zeroes of the polynomial,

α +β+γ = 2+1+1 = 4 = -(-4)/1 = –b/a

αβ+βγ+γα = 2×1+1×1+1×2 = 5 = 5/1= c/a

αβγ = 2×1×1 = 2 = -(-2)/1 = -d/a

Hence, the relationship between the zeroes and the coefficients are satisfied.

**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.**

**Solution:**

Let us consider the cubic polynomial is ax^{3}+bx^{2}+cx+d and the values of the zeroes of the polynomials be α, β, γ.

As per the given question,

α+β+γ = -b/a = 2/1

αβ +βγ+γα = c/a = -7/1

α βγ = -d/a = -14/1

Thus, from above three expressions we get the values of coefficient of polynomial.

a = 1, b = -2, c = -7, d = 14

Hence, the cubic polynomial is x^{3}-2x^{2}-7x+14

**3. If the zeroes of the polynomial x**^{3}**-3x**^{2}**+x+1** **are a – b, a, a + b, find a and b.**

**Solution:**

We are given with the polynomial here,

p(x) = x^{3}-3x^{2}+x+1

And zeroes are given as a – b, a, a + b

Now, comparing the given polynomial with general expression, we get;

∴px^{3}+qx^{2}+rx+s = x^{3}-3x^{2}+x+1

p = 1, q = -3, r = 1 and s = 1

Sum of zeroes = a – b + a + a + b

-q/p = 3a

Putting the values q and p.

-(-3)/1 = 3a

a=1

Thus, the zeroes are 1-b, 1, 1+b.

Now, product of zeroes = 1(1-b)(1+b)

-s/p = 1-b^{2}

-1/1 = 1-b^{2}

b^{2} = 1+1 = 2

b = ±√2

Hence,1-√2, 1 ,1+√2 are the zeroes of x^{3}-3x^{2}+x+1.

**4. If two zeroes of the polynomial x**^{4}**-6x**^{3}**-26x**^{2}**+138x-35** **are 2 ±**√**3,** **find other zeroes.**

**Solution:**

Since this is a polynomial equation of degree 4, hence there will be total 4 roots.

Let f(x) = x^{4}-6x^{3}-26x^{2}+138x-35

Since 2 +√**3 **and 2-√**3 **are zeroes of given polynomial f(x).

∴ [x−(2+√**3**)] [x−(2-√**3)**] = 0

(x−2−√**3**)(x−2+√**3**) = 0

On multiplying the above equation we get,

x^{2}-4x+1, this is a factor of a given polynomial f(x).

Now, if we will divide f(x) by g(x), the quotient will also be a factor of f(x) and the remainder will be 0.

So, x^{4}-6x^{3}-26x^{2}+138x-35 = (x^{2}-4x+1)(x^{2} –2x−35)

Now, on further factorizing (x^{2}–2x−35) we get,

x^{2}**– (7−5)x −35** = x^{2}– 7x+5x+35 = 0

x(x −7)+5(x−7) = 0

**(x+5) (x−7) = 0**

So, its zeroes are given by:

x= −5 and x = 7.

Therefore, all four zeroes of given polynomial equation are: 2+√**3**, 2-√**3**, **−5 and 7.**

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