Question

Select the correct answer from each drop-down menu. Consider the following polynomials equations.

3x^2(x-1) and -3x^3+ 4x^2-2x+1


Perform each operation and determine if the result is a polynomial.


Is the result of A + B a polynomial? yes or no


Is the result of A - B a polynomial? yes or no


Is the result of A · B a polynomial? yes or no

Answer 1

Answer:

Yes to all 3 questions.

Step-by-step explanation:

3x^2(x-1)  +  -3x^3+ 4x^2-2x+1

= 3x^3 - 3x^2  - 3x^3 + 4x^2 - 2x + 1

=  x^2 - 2x + 1,

So A + B is a polynomial.

3x^2(x-1)  -  (-3x^3+ 4x^2-2x+1)

= 3x^3 - 3x^2 + 3x^3 - 4x^2 + 2x - 1

= 6x^3 - 7x^2 + 2x - 1.

So A - B is also a polynomial.

3x^2(x-1)( -3x^3+ 4x^2-2x+1)

= (3x^3 - 3x^2) ( -3x^3 + 4x^2 - 2x + 1)

= -9x^6 + 12x^5 - 6x^4 + 3x^3  + 9x^5 - 12x^4 + 6x^3 - 3x^2

= -9x^6 + 21x^5 - 18x^4 + 9x^3 - 3x^2.

So A * B is a polynomial.

Answer 2

The result of the product, sum and difference of the two polynomials are all polynomials.

Given the polynomials :

• A = 3x²(x-1) = 3x³ - 3x²

• B = -3x³ + 4x² -2x + 1

The sum of the polynomial :

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

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

A + B = x² - 2x + 1

Yes, A + B is a polynomial

2.) The difference of the polynomial :

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

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

A - B = 6x³ - 7x² + 2x - 1

Yes, A - B is a polynomial

3.) The product of the polynomial :

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

A × B = -9x^6 + 12x^5 - 6x^4 + 3x^3 + 9x^5 - 12x^4 + 6x^3 - 3x^2

A × B = - 9x^6 + 19x^5 - 18x^4 + 9x^3 - 3x^2

Yes, A × B is a polynomial

Therefore, the sum, difference and product of polynomials A and B are all polynomials.

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