**Previous exercise – Polynomials – Exercise 8.1 – Class X**

#### Polynomials – Exercise 8.2

- Divide p(x) by g(x) in each o the following cases and verify division algorithm

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

(ii)p(x) = x^{2} – 9x + 9, g(x) = x – 3

(iii) p(x) = x^{3} + 4x^{2} -5x + 6, g(x) = x + 1

(iv) p(x) = x^{4} – 3x^{2} – 4, g(x) = x + 2

(v) p(x) = x^{3} – 1, g(x) = x – 1

(vi) p(x) = x^{4} – 4x^{2}+ 12x + 9, g(x) = x^{2} + 2x – 3

- Find the divisor g(x) , when the polynomial p(x) = 4x
^{3}+ 2x^{2}– 10x +2 is divided by g(x) and the quotient and the remainder obtained are (2x^{2}+4x + 1) and 5 respectively. - On dividng the polynomial p(x) = 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) - A polynomial p(x) id divided by g(x), the obtained quotient q(x) and the remainder are given in the table. Find the p(x) in each case.

Sl | p(x) | g(x) | q(x) | r(x) |

i | ? | x – 2 | x^{2} – x + 1 | 4 |

ii | ? | x + 3 | 2x^{2} + x + 5 | 3x +1 |

iii | ? | 2x + 1 | x^{3} + 3x^{2} – x +1 | 0 |

iv | ? | x – 1 | x^{3} – x^{2} – x – 1 | 2x – 4 |

v | ? | x^{2} + 2x + 1 | x^{4} – 2x^{2} + 5x – 7 | 4x + 12 |

- Find the quotient and remainder on dividing p(x) by g(x) in each of the fooling cases, without actual division

(i) p(x) = x^{2} + 7x + 10; g(x) = x – 2

(ii) p(x) = x^{3 }+4x^{2} – 6x + 2; g(x) = x – 3

- What must be subtracted from (x
^{3}+ 5x^{2}+ 5x + 8) so that the resulting polynomial exactly divisible by (x^{2}+ 3x – 2)? - What should be added to (x
^{4}– 1) so that it is exactly divisible by (x^{2}+ 2x + 1)?

#### Polynomials – Exercise 8.2 – Solutions:

**Divide p(x) by g(x) in each o the following cases and verify division algorithm**

**(i)p(x) = x ^{2} + 4x + 4, g(x) = x + 2**

Solution:

Given, we have to divide p(x) by g(x) i.e., we have to divide x^{2} + 4x + 4 by x + 2.

Verifying:

Quotient, q(x) = (x + 2)

Remainder, r(x) = 0

By division algorithm for polynomials, p(x) = [g(x) * q(x)] + r(x)

Here, p(x) = x^{2} + 4x + 4 ;

g(x) = x + 2 ;

q(x) = x + 2 ;

r(x) = 0

⸫ p(x) = [(x + 2) * (x + 2)] + 0

= [x^{2} + 2x + 2x + 4] + 0

= x^{2} + 4x + 4 = p(x)

⸫ division algorithm is verified.

**(ii)p(x) = x ^{2} – 9x + 9, g(x) = x – 3**

Solution:

Given, we have to divide p(x) by g(x) i.e., we have to divide x^{2} – 9x + 9 by x – 3 .

Verifying:

Quotient, q(x) = (x – 6)

Remainder, r(x) = 27

By division algorithm for polynomials, p(x) = [g(x) * q(x)] + r(x)

Here, p(x) = x^{2} – 9x + 9 ;

g(x) = x – 3 ;

q(x) = x – 6 ;

r(x) = 27

⸫ p(x) = [(x – 3) * (x – 6)] + 27

= [x^{2} – 6x – 3x – 18] + 27

= x^{2} – 9x – 18 + 27

=x^{2} – 9x + 9 = p(x)

⸫ division algorithm is verified.

**(iii) p(x) = x ^{3} + 4x^{2} -5x + 6, g(x) = x + 1**

Solution:

Given, we have to divide p(x) by g(x) i.e., we have to divide x^{3} + 4x^{2} -5x + 6 by x + 1.

Verifying:

Quotient, q(x) = (x^{2} + 2x – 7)

Remainder, r(x) = 13

By division algorithm for polynomials, p(x) = [g(x) * q(x)] + r(x)

Here, p(x) = x^{3} + 4x^{2} – 5x + 6 ;

g(x) = x + 1 ;

q(x) = x^{2} + 3x – 8 ;

r(x) = 14

⸫ p(x) = [(x^{2} + 3x – 8) * (x + 1)] + 14

= x^{3} + 3x^{2} – 8x + x^{2} + 3x – 8 + 14

= x^{3} + 4x^{2} – 5x + 6

= p(x)

⸫ division algorithm is verified.

**(iv) p(x) = x ^{4} – 3x^{2} – 4, g(x) = x + 2**

Solution:

Given, we have to divide p(x) by g(x) i.e., we have to divide x^{4} – 3x^{2} – 4 by x + 2.

Verifying:

Quotient, q(x) = (x^{3} – 2x^{2} + x – 2)

Remainder, r(x) = 0

By division algorithm for polynomials, p(x) = [g(x) * q(x)] + r(x)

Here, p(x) = x^{4} – 3x^{2} – 4;

g(x) = x + 2 ;

q(x) = x^{3} – 2x^{2} + x – 2;

r(x) = 0

⸫ p(x) = [(x + 2) * (x^{3} – 2x^{2} + x – 2)] + 0

= [x^{4} – 2x^{3} + x^{2} – 2x + 2x^{3} – 4x^{2} + 2x – 4 ] + 0

= x^{4} + 0 – 3x^{2} + 0 – 4

x^{4} – 3x^{2} – 4 = p(x)

⸫ division algorithm is verified.

**(v) p(x) = x ^{3} – 1, g(x) = x – 1**

Solution:

Given, we have to divide p(x) by g(x) i.e., we have to divide x^{3} – 1 by x – 1.

Verifying:

Quotient, q(x) = (x^{2} + x + 1)

Remainder, r(x) = 0

By division algorithm for polynomials, p(x) = [g(x) * q(x)] + r(x)

Here, p(x) = x^{3} – 1 ;

g(x) = x – 1 ;

q(x) = x^{2} + x + 1;

r(x) = 0

⸫ p(x) = [(x – 1) * (x^{2} + x + 1)] + 0

= [x^{3} + x^{2} + x – x^{2} – x – 1 ] + 0

= x^{3} + 0 – 0 – 1

= x^{3} – 1 = p(x)

⸫ division algorithm is verified.

**(vi) p(x) = x ^{4} – 4x^{2}+ 12x + 9, g(x) = x^{2} + 2x – 3**

Solution:

Given, we have to divide p(x) by g(x) i.e., we have to divide x^{4} – 4x^{2}+ 12x + 9 by x^{2} + 2x – 3.

Verifying:

Quotient, q(x) = (x^{2} – 2x + 3)

Remainder, r(x) = 0

By division algorithm for polynomials, p(x) = [g(x) * q(x)] + r(x)

Here, p(x) = x^{4} – 4x^{2}+ 12x + 9;

g(x) = x^{2} + 2x – 3;

q(x) = x^{2} – 2x + 3;

r(x) = 18

⸫ p(x) = [(x^{2} – 2x + 3) * (x^{2} + 2x – 3)] + 18

= [x^{4} +2x^{3} – 3x^{2} – 2x^{3} – 4x^{2} + 6x + 3x^{2} + 6x – 9 ] + 18

= x^{4} + 0 – 4x^{2} + 12x + 9

= x^{4} – 4x^{2} + 12x + 9

= p(x)

⸫ division algorithm is verified.

**Find the divisor g(x) , when the polynomial p(x) = 4x**^{3}+ 2x^{2}– 10x +2 is divided by g(x) and the quotient and the remainder obtained are (2x^{2}+4x + 1) and 5 respectively.

Solution:

By division algorithm for polynomials, p(x) = [g(x) * q(x)] + r(x)

p(x) = 4x^{3} + 2x^{2} – 10x +2

q(x) = 2x^{2} +4x + 1

r(x) = 5

We have to find g(x),

p(x) = [g(x) * q(x)] + r(x)

g(x) = ^{p(x) – r(x)}/_{q(x)}

= ^{(4x^3 + 2x^2 – 10x +2) – (5)}/_{2x^2 +4x + 1}

= ^{4x^3 + 2x^2 – 10x +2 – 5}/_{2x^2 +4x + 1}

= ^{4x^3 + 2x^2 – 10x – 3}/_{2x^2 +4x + 1}

g(x) = 2x – 2

**On dividing the polynomial p(x) = 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:

By division algorithm for polynomials, p(x) = [g(x) * q(x)] + r(x)

p(x) = x^{3} – 3x^{2} + x + 2

q(x) = (x – 2)

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

We have to find g(x),

p(x) = [g(x) * q(x)] + r(x)

g(x) = ^{p(x) – r(x)}/_{q(x)}

= ^{(x^3 – 3x^2 + x + 2)-(-2x + 4)}/_{x – 2}

= ^{x^3 – 3x^2 + x + 2 +2x – 4}/_{x – 2}

=^{ x^3 – 3x^2 + 3x – 2}/_{x – 2}

g(x) = x^{2} – x + 1

**A polynomial p(x) id divided by g(x), the obtained quotient q(x) and the remainder are given in the table. Find the p(x) in each case.**

Sl | p(x) | g(x) | q(x) | r(x) |

i | ? | x – 2 | x^{2} – x + 1 | 4 |

ii | ? | x + 3 | 2x^{2} + x + 5 | 3x +1 |

iii | ? | 2x + 1 | x^{3} + 3x^{2} – x +1 | 0 |

iv | ? | x – 1 | x^{3} – x^{2} – x – 1 | 2x – 4 |

v | ? | x^{2} + 2x + 1 | x^{4} – 2x^{2} + 5x – 7 | 4x + 12 |

Solution:

(i) By division algorithm for polynomials, p(x) = [g(x) * q(x)] + r(x)

g(x) = x – 2 ;

q(x) = x^{2} – x + 1 ;

r(x) = 4 ;

p(x) = [(x – 2)*(x^{2} – x + 1)] + 4

= x^{3} – x^{2} + x – 2x^{2} + 2x – 2 + 4

= x^{3} – 3x^{2} + 3x + 2

(ii) By division algorithm for polynomials, p(x) = [g(x) * q(x)] + r(x)

g(x) = x + 3

q(x) = 2x^{2} + x + 5

r(x) = 3x + 1

p(x) = [(x + 3)*(2x^{2} + x + 5)] + (3x + 1)

= 2x^{3} + x^{2} + 5x + 6x^{2} + 3x + 15 + 3x + 1

= 2x^{3} + 7x^{2} + 11x + 16

(iii) By division algorithm for polynomials, p(x) = [g(x) * q(x)] + r(x)

g(x) = 2x + 1

q(x) = x^{3} + 3x^{2} – x +1

r(x) = 0

p(x) = (2x + 1)(x^{3} + 3x^{2} – x + 1) + 0

= 2x^{4} + 6x^{3} – 2x^{2} + 2x + x^{3} + 3x^{2} – x + 1

= 2x^{4} + 7x^{3} + x^{2} + x + 1

(iv) By division algorithm for polynomials, p(x) = [g(x) * q(x)] + r(x)

g(x) = x – 1

q(x) = x^{3} – x^{2} – x – 1

r(x) = 2x – 4

p(x) = (x^{3} – x^{2} – x – 1)*(x – 1) + (2x – 4)

= x^{4} – x^{3} – x^{2} – x – x^{3} + x^{2} + x + 1 + 2x – 4

= x^{4} – 2x^{3} + 2x – 3

(v) By division algorithm for polynomials, p(x) = [g(x) * q(x)] + r(x)

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

q(x) = x^{4} – 2x^{2} + 5x – 7

r(x) = 4x + 12

p(x) = (x^{2} + 2x + 1)( x^{4} – 2x^{2} + 5x – 7) + 4x + 12

= x^{6} – 2x^{4} + 5x^{3} – 7x^{2} +2x^{5} – 4x^{3} + 10x^{2} – 14x + x^{4} – 2x^{2} + 5x – 7 + 4x + 12

p(x) = x^{6} + 2x^{5 }– x^{4} + x^{3} + x^{2} – 5x + 5

**Find the quotient and remainder on dividing p(x) by g(x) in each of the following cases, without actual division**

**(i) p(x) = x ^{2} + 7x + 10; g(x) = x – 2**

Solution:

p(x) = x^{2} + 7x + 10

⸫degree of p(x) = 2

g(x) = x – 2

⸫degree of g(x) = 1

⸫degree of quotient q(x) = 2 – 1 = 1 and degree of remainder r(x) is 0.

Let q(x) = ax + b (polynomial of degree 1) and remainder, r(x) = k

By division algorithm for polynomials, p(x) = [g(x) * q(x)] + r(x)

x^{2} + 7x + 10 = (x – 2)*(ax + b) + k

x^{2} + 7x + 10 = ax^{2} + bx – 2ax – 2b + k

x^{2} + 7x + 10 = ax^{2} + x (b – 2a) – 2b + k

Let us compare the coefficients of x^{2}, x and k to get the values of a, b, and k

⸫ a = 1 ; coefficients of x^{2} on both the sides

⸫ b – 2a = 7 ; coefficients of x on both the sides

⸫ 10 = -2b + k ; constants on both the sides

We have to solve these equations to get the value of a, b and k

Since a = 1

b – 2a = 7

⇒ b = 7 + 2a = 7 + 2(1) = 9

10 = -2b + k

k = 10 + 2b = 10 + 9×2 = 10 + 18 = 28

Since q(x) = ax + b = x + 9

r(x) = 28

Therefore, quotient = x + 9 and remainder 28.

**(ii) p(x) = x ^{3 }+4x^{2} – 6x + 2; g(x) = x – 3**

Solution:

p(x) = x^{3 }+4x^{2} – 6x + 2

⸫degree of p(x) = 3

g(x) = x –3

⸫degree of g(x) = 1

⸫degree of quotient q(x) = 3 – 1 = 2 and degree of remainder r(x) is 0.

Let q(x) = ax^{2} + bx + c (polynomial of degree 1) and remainder, r(x) = k

By division algorithm for polynomials, p(x) = [g(x) * q(x)] + r(x)

x^{3 }+4x^{2} – 6x + 2= (x – 3)*(ax^{2} + bx + c) + k

x^{3 }+4x^{2} – 6x + 2 = ax^{3} + bx^{2} + cx – 3ax^{2} – 3bx – 3c + k

x^{3 }+4x^{2} – 6x + 2 = ax^{3} +x^{2}(b – 3a)+x (c – 3b) – 3c + k

Let us compare the coefficients of x^{3}, x^{2}, x and k to get the values of a, b, c and k

⸫ a = 1 ; coefficients of x^{3} on both the sides

⸫ b – 3a = 4 ; coefficients of x^{2} on both the sides

⸫ -6 = c – 3b ; coefficients of x on both the sides

⸫ 2 = -3c + k ; constants on both the sides

Solve these equations to get the value of a, b and k

Since a = 1

b – 3a = 4

⇒ b = 4 + 3a = 4 + 3 = 7

b = 7

-6 = c – 3b

⇒ – 6 = c – 3(7)

⇒ – 6 = c – 21

⇒ c = -6 + 21 = 15

c = 15

2 = -3c + k

k = 2 + 3c = 2 + 3×15 = 2 + 45 = 47

Since q(x) = ax^{2} + bx + c = x^{2} + 7x + 15

r(x) = 47

Therefore, quotient = x^{2} + 7x + 15 and remainder 47.

**What must be subtracted from (x**^{3}+ 5x^{2}+ 5x + 8) so that the resulting polynomial exactly divisible by (x^{2}+ 3x – 2)?

Solution:

To find what must be subtracted from (x^{3} + 5x^{2} + 5x + 8) so that the resulting polynomial exactly divisible by (x^{2} + 3x – 2), we need to divide x^{3} + 5x^{2} + 5x + 8 by x^{2} + 3x – 2

On dividing x^{3} + 5x^{2} + 5x + 8 by x^{2} + 3x – 2, we get quotient q(x) = (x +2) and the remainder r(x) = (-x + 8).

Therefore, we must subtract (-x + 8) from (x^{3} + 5x^{2} + 5x + 8) so that the resulting polynomial exactly divisible by (x^{2} + 3x – 2).

**What should be added to (x**^{4}– 1) so that it is exactly divisible by (x^{2}+ 2x + 1)?

Solution:

To find what should be added to (x^{4} – 1) so that it is exactly divisible by (x^{2} + 2x + 1), we need to divide x^{4} – 1 from x^{2} + 2x + 1

On dividing x^{4} – 1 by x^{2} + 2x + 1, we get quotient q(x) = (x^{2} – 2x + 3) and the remainder r(x) = (-4x – 4).

Therefore, we must add (4x + 4) from (x^{4} – 1) so that the resulting polynomial exactly divisible by (x^{2} + 2x + 1).

**Next Exercise – Polynomials – Exercise 8.3 – Class X**