Respuesta :

Matheng
The polynomial function has the form :
[tex]f(x) = a_{n} x^{n} +a_{n-1} x^{n-1} +a_{n-2} x^{n-2}+ ..............+ a_{1} x+a_{ 0 } [/tex]

A = 3x²(x-1) = 3x³ - 3x²
B = -3x³ + 4x² -2x + 1

A + B = (3x³ - 3x²) + (-3x³ + 4x² -2x + 1) = x² - 2x + 1
∴ Yes, The result of (A+B) is polynomial


A - B = (3x³ - 3x²) - (-3x³ + 4x² -2x + 1) = 6x³ - 7x² + 2x - 1
∴ Yes, The result of (A-B) is polynomial 

A * B = (3x³ - 3x²) * (-3x³ + 4x² -2x + 1)
∴ Yes, The result of (A-B) is polynomial

Answer with explanation:

An expression of the form [tex]A_{0}x^n+A_{1}x^{n-1}+.............+A_{n}[/tex] is called a Polynomial,if it consist of only one variable in entire terms, and degree of each variable is non negative integer.

The given Polynomial  are

[tex]A=3x^2(x-1)\\\\B= -3 x^3 + 4 x^2 - 2 x +1[/tex]

[tex]1. A +B=3x^2(x-1) + [-3 x^3 + 4 x^2 - 2 x +1]\\\\=3x^3- 3 x^2 -3x^3+4 x^2- 2 x + 1\\\\{\text{Adding and subtracting like terms}}\\\\=x^2-2 x +1\\\\2. A - B=3x^2(x-1) - [-3 x^3 + 4 x^2 - 2 x +1]\\\\=3x^3- 3 x^2 +3x^3-4 x^2+ 2 x - 1\\\\{\text{Adding and subtracting like terms}}\\\\=6 x^3-7 x^2+2 x -1\\\\3. A \times B=[3x^2(x-1)] \times [-3 x^3 + 4 x^2 - 2 x +1]\\\\=[3 x^3-3 x^2]\times [-3 x^3 + 4 x^2 - 2 x +1]\\\\3 x^3\times [-3 x^3 + 4 x^2 - 2 x +1]-3 x^2\times [-3 x^3 + 4 x^2 - 2 x +1][/tex]

[tex]=-9x ^6+12 x^5-6 x^4+3 x^3+9 x^5-12 x^4+6 x^3-3 x^2\\\\{\text{Adding and subtracting like terms and used property of exponents}}\rightarrow x^a\times x^b=x^{a+b}\\\\-9 x^6+21 x^5-18 x^4+9 x^3-3 x^2[/tex]

→→→All the three, that is

1.A +B

2. A -B

3. A× B

are Polynomials

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