Question

Use the functions f(x)=7x3−3x2−x+14 f x = 7 x 3 - 3 x 2 - x + 14 and g(x)=2x3+2x+5 g x = 2 x 3 + 2 x + 5 as examples to show that the set of polynomials is closed under subtraction.

Answer 1

The given set of polynomials f(x) and g(x) are closed under subtraction.

Explanation:

Given that the functions f ad g are defined by

`f(x)=7x^3-3x^2-x+14` and
`g(x)=2x^3+2x+5` respectively.

To show that the set of polynomials is closed under subtraction :

Now subtract the given polynomials

`f(x)-g(x)=(f-g)(x)`

`=7x^3-3x^2-x+14-(2x^3+2x+5)`

`=7x^3-3x^2-x+14-2x^3-2x-5`

`=5x^3-3x^2-3x+9`

∴
`f(x)-g(x)=5x^3-3x^2-3x+9`

• When subtracting the polynomials the variables and their exponents remains same only variation in their coefficients.

Hence the given polynomials f(x) and g(x) are closed under subtraction.

∴
`f(x)-g(x)=5x^3-3x^2-3x+9` are closed under subtraction,

Hence showed.