Question

Consider the following polynomials equations
A= 3x^2 (x-1)
B= -3x^3 + 4x^2 - 2x +1

Answer 1

Answer with explanation:

An expression of the form
`A_(0)x^n+A_(1)x^(n-1)+.............+A_(n)` 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

`A=3x^2(x-1)\\\\B= -3 x^3 + 4 x^2 - 2 x +1`

`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 * B=[3x^2(x-1)] * [-3 x^3 + 4 x^2 - 2 x +1]\\\\=[3 x^3-3 x^2]* [-3 x^3 + 4 x^2 - 2 x +1]\\\\3 x^3* [-3 x^3 + 4 x^2 - 2 x +1]-3 x^2* [-3 x^3 + 4 x^2 - 2 x +1]`

`=-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* x^b=x^(a+b)\\\\-9 x^6+21 x^5-18 x^4+9 x^3-3 x^2`

→→→All the three, that is

1.A +B

2. A -B

3. A× B

are Polynomials

Answer 2

The polynomial function has the form :

`f(x) = a_(n) x^(n) +a_(n-1) x^(n-1) +a_(n-2) x^(n-2)+ ..............+ a_(1) x+a_( 0 )`

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