SUBTRACTION OF POLYNOMIALS

Question 1 :

Subtract the second polynomial from the first polynomial and find the degree of the resultant polynomial

(i)  p(x)  =  7x² + 6x - 1 and q(x)  =  6x - 9 

Solution :

p(x) - q(x)  =  (7x² + 6x - 1) - (6x - 9)

  =  7x² + 6x - 1 - 6x + 9

  =  7x² + 6x - 6x  - 1 + 9

  =  7x² + 8

Degree of the resultant polynomial is 2.

(ii)  f(y)  =  6y² - 7y + 2 and g(y)  =  7y + y³

Solution :

f(y) - g(y)  =  (6y² - 7y + 2) - (7y + y³)

  =  6y² - 7y + 2 - 7y - y³

  =  - y³ + 6y² - 7y - 7y + 2

  =  - y³ + 6y² - 14y + 2

Degree of the resultant polynomial is 3.

(iii)  h(z)  =  z⁵ - 6z⁴ + z and f(z)  =  6z² + 10z - 7

Solution :

h(z) - f(z)  =  (z⁵ - 6z⁴ + z) - (6z² + 10z - 7)

  =  z⁵ - 6z⁴ + z - 6z² - 10z + 7

  =  z⁵ - 6z⁴ - 6z² - 9z + 7

Degree of the resultant polynomial is 5.

Question 2 :

What should be added to 2x³ + 6x² - 5x + 8 to get 3x³ - 2x² + 6x + 15 ?

Solution :

Let p(x) be the required polynomial to be added

By adding p(x) and 2x³ + 6x² - 5x + 8, we will get 3x³ - 2x² + 6x + 15

p(x) + (2x³ + 6x² - 5x + 8)  =  3x³ - 2x² + 6x + 15

p(x)  =  (3x³ - 2x² + 6x + 15) - (2x³ + 6x² - 5x + 8)

p(x)  =  3x³ - 2x³- 2x² + 6x²  + 6x - 5x + 15 + 8

p(x)  =  x³ + 4x² + x + 23

Question 3 :

What must be subtracted from 2x⁴ + 4x² - 3x + 7 to get 3x³ - x² + 2x + 1?

Solution :

Let p(x) be the required polynomial to be subtracted.

 (2x⁴ + 4x² - 3x + 7) - p(x)  =  3x³ - x² + 2x + 1

p(x)  =  (2x⁴ + 4x² - 3x + 7) - (3x³ - x² + 2x + 1)

  =  2x⁴ + 4x² - 3x + 7 - 3x³ + x² - 2x - 1

  =  2x⁴ - 3x³ + 4x² + x² - 3x - 2x + 7 - 1

p(x)  =  2x⁴ - 3x³ + 5x² - 5x + 6

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