WORKSHEET ON OPERATIONS WITH POLYNOMIALS

Practice Questions on Adding Polynomials

Add the following polynomials

1)  (7j3 – 2) + (5j3 – j – 3)

2)  (8a5 – 4 ) + (3a5 + a – 2)

3)  (6m5 + 1) + (2m5 + 9m – 1)

4)  (3m5 + 1) + (9m5 + 3m – 2)

5)  (-5x2 – x + 4) + (-3x2 – 5x + 2)

Solution

6)  Add ( 2x³ + 5x² - 2x + 7 ) and ( x³ + 4x² - x + 6)

7)  Add ( 3x³ - 2x² - x + 4 ) and ( 2x³ + 7x² - 3x - 3 )

8)  Add  2( x³ - x² + 6x - 2 ) and ( 5x⁶ + 7x⁵ - 3x - 3 )

9)  Add -1( x⁶ + x³ + 6x² - 2 ) and 2( 5x⁶ + 7x⁵ - 3x - 3 )

10)  Add 5( 5x⁶ + 2x³ - 6x² - 2 ) + 6(-3x⁶ + 2x⁵ + 2x + 1 )

Solution

Answers :

1)  12j3 – j – 5

2)  11a5 + a - 6

3)  8m5 + 9m

4)  12m5 + 3m - 1

5)  -8x2 - 6x + 6

6)  3x3 + 9x2 - 3x + 13

7)  5x³ + 5x² - 4x + 1

8)   5x⁶ + 7x⁵ + 2x³ - 2x² + 9x - 7

9)  9x⁶ + 14x⁵ - x³ - 6x² - 6x - 4

10)  7x⁶ + 12x⁵ + 10x³ - 30x² + 12x - 4

Practice Questions on Subtracting  Polynomials

Subtract the following polynomials

1)  Subtract 5ab from 8ab.

2)  Subtract (4x + 5y2)  from  (5x - y2)

3)  Subtract (2x2 + 2y2 - 6)  from  (3x2 - 7y2 + 9)

4)  Subtract (x3 + 4x2 - x + 6) from (2x3 + 5x2 - 2x + 7)

5)  Subtract (3 + 3x - 7x5 - 5x6) from 2(x3 - x2 + 6x - 2)

Solution

6)  Subtract 2x³ + 5x² - 2x - 11 from 3x³ - 2x² - 5x - 6

7)  Subtract x³ + 4x² - 12x - 5  from 5x³ + 3x² + 2x - 10

8) Subtract 12x³ + 14x² + 17x - 12 from 15x³+22x²+17x-19 

9)  Subtract 5x³ + 3x² + 7x - 6 from 3x³ + 2x² + 6x - 4

10)  Subtract x³ + 32x² + 17x - 16 from 13x³+23x²+16x-14

Solution

Answers :

1)  3ab

2)  x - 6y2

3)  x2 - 9y2 + 15

4)  x3 + x2 - x + 1

5) 5x6 + 7x5 + 2x3 - 2x2 + 9x - 7

6)  x³ - 7x² - 3x + 5

7)  4x³ - x² + 14x - 5

8)  3x³ + 8x² - 7

9)  -2x³ - x² - x + 2

10)  12x³ - 9x² - x + 2

Practice Questions on Multiplying  Polynomials

Multiply the following polynomials

1)  (4x2)(5x3)

2)  (a2b2c3)(abc2)

3)  (0.5a3b)(a2c2)(6b2)

4)  2(3x2 + 5x + 7)

5)  3ab(5a2 + b)

6)  2xy(x2y + xz - xy)

7)  (x + 3)(x - 6)

8)  (x + 2)2

9)  (3a2 - b)(a2 - 2b)

10)  (x + 3)(x2 - 5x + 7)

11)  (a + b)(2a2 - 5ab + 3b2)

12)  (2x + 3y)(x2 - xy + y2)

Solution

Answers :

1)  20x5

2)  a3b3c5

3)  3a5b3c2

4)  6x2 + 10x + 14

5)   15a3b + 3ab2

6)  2x3y2 + 2x2yz - 2x2y2

7)  x2 - 3x - 18

8)  x2 + 4x + 4

9)  3a4 - 7a2b + 2b2

10)  x3 - 2x2 - 8x + 21

11)  2a3 - 3a2b - 2ab2 + 3b3

12)  2x3 + x2y - xy2 + 3y3

Practice Questions on Dividing  Polynomials using Synthetic Division

Find the quotient and remainder for the following polynomials using synthetic division

1)  (x3 + x2 - 3 x + 5 ) ÷ ( x - 1 ) 

2)  (3 x³ - 2 x² + 7 x - 5) ÷ (x + 3)

Solution

3)  (x3 + x2 - 7x - 3) ÷ (x - 3)

4)  (x3 + 2x2 - x - 4) ÷ (x + 2)

5)  (3x3 - 2x2 + 7x - 5) ÷ (x + 3)

6)  (8x4 - 2x2 + 6x + 5) ÷ (4x + 1)

7)   (8x4 - 2x2 + 6x + 5) ÷ (4x + 1)

8)  If the quotient obtained on dividing (8x4 - 2x2 + 6x - 7) by (2x + 1) is (4x3 + px2 - qx + 3) then find p, q and also the remainder.

9)  If the quotient obtained on dividing 3x3 + 11x2 + 34x + 106 by x - 3 is 3x2 + ax + b, then find a, b and also the remainder.

Solution

Answers :

1)  Quotient = x² + 2 x - 1, Remainder = 4

2)  Quotient = 3 x² - 11 x + 40, Remainder = - 125

3)  Quotient  =  x2 + 4x - 3, Remainder  =  -12

4)  Quotient  =  x2 + 0x - 1, Remainder  =  -2

5)  Quotient  =  3x2 - 11x + 40, Remainder  =  -125

6)  Quotient  =  8x2 - x + 40, Remainder  =  -125

7)  Quotient : 8x3 - 2x2 -(3x/2) + (51/8), Remainder  : 109/32

8)  p  =  -2 and q  =  0, Remainder  =  -10

9)  a  =  20 and b  =  94, Remainder  =  388

Practice Questions on Dividing  Polynomials using Long Division

Find the quotient and remainder for the following polynomials using long division

1)  Find the quotient and the remainder when (10- 4x + 3x²) is divided by x - 2.

2)  Find the quotient and the remainder when (4x³ + 6x² - 23 x - 15) is divided by 3 + x

Solution

3)  (10 + 7x + x2) ÷ (x + 2)

4)  (3x3 + 11x2 + 9x - 5) ÷ (x + 2)

5)  (x2 - 25) ÷ (x + 5)

6)  (x2 - 1000) ÷ (x - 100)

7)  (30x3 + 50x) ÷ 5x

8)  Find a and b if the polynomial x5 - ax + b is divisible by   x2 - 4

9)  The volume of a rectangular solid is given by the polynomial 3x4 - 3x3 - 33x2 + 54x. The length of the solid is given by 3x and the width is given by x - 2. Find the height of the solid.

10)  The area of a rectangle is given by 3x3 + 14x2 - 23x + 6. The width of the rectangle is given by x + 6. Find an expression for the length of the rectangle.

Solution

Answers :

1)  Quotient  =  3x + 2, Remainder  =  14

2)  Quotient  =  4 x²-6 x - 5, Remainder = 0

3)  Quotient  =  x + 5, Remainder  =  0

4)  Quotient  =  3x2 + 5x - 1, Remainder  =  - 3

5)  Quotient  =  x - 5, Remainder  =  0

6)  Quotient  =  x + 100, Remainder  =  0

7)  Quotient  =  6x2 + 10, Remainder  =  0

8)   a  =  16 and b  =  0

9)  Height of the solid is x2 + x - 9

10)  Length of the rectangle is 3x2 - 4x + 1

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