Question

select the correct answer from each drop-down menu.
consider the following polynomials equations.
$a = 3x^{2}(x - 1)$
$b = -3x^{3} + 4x^{2} - 2x + 1$
perform each operation and determine if the result is a polynomial.
is the result of $a + b$ a polynomial?
is the result of $a - b$ a polynomial?
is the result of $a \bullet b$ a polynomial?

Explanation:

Step1: Expand polynomial A

$A = 3x^2(x-1) = 3x^3 - 3x^2$

Step2: Calculate A + B

$A+B=(3x^3 - 3x^2)+(-3x^3 + 4x^2 - 2x + 1)=x^2 - 2x + 1$
This is a polynomial (sum of polynomials is a polynomial).

Step3: Calculate A - B

$A-B=(3x^3 - 3x^2)-(-3x^3 + 4x^2 - 2x + 1)=3x^3 -3x^2 +3x^3 -4x^2 +2x -1=6x^3 -7x^2 +2x -1$
This is a polynomial (difference of polynomials is a polynomial).

Step4: Calculate A • B

$A\cdot B=(3x^3 - 3x^2)(-3x^3 + 4x^2 - 2x + 1)$
$=3x^3(-3x^3 + 4x^2 - 2x + 1) -3x^2(-3x^3 + 4x^2 - 2x + 1)$
$=-9x^6 +12x^5 -6x^4 +3x^3 +9x^5 -12x^4 +6x^3 -3x^2$
$=-9x^6 +21x^5 -18x^4 +9x^3 -3x^2$
This is a polynomial (product of polynomials is a polynomial).

Answer:

Is the result of $A + B$ a polynomial? Yes
Is the result of $A - B$ a polynomial? Yes
Is the result of $A \bullet B$ a polynomial? Yes