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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.

Respuesta :

Answer:

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

Step-by-step explanation:

Given that the functions f ad g are defined by

[tex]f(x)=7x^3-3x^2-x+14[/tex] and [tex]g(x)=2x^3+2x+5[/tex] respectively.

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

Now subtract the given polynomials

[tex]f(x)-g(x)=(f-g)(x)[/tex]

[tex]=7x^3-3x^2-x+14-(2x^3+2x+5)[/tex]

[tex]=7x^3-3x^2-x+14-2x^3-2x-5[/tex]

[tex]=5x^3-3x^2-3x+9[/tex]

∴ [tex]f(x)-g(x)=5x^3-3x^2-3x+9[/tex]

  • 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.

∴ [tex]f(x)-g(x)=5x^3-3x^2-3x+9[/tex] are closed under subtraction,

Hence showed.

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