Question

select the correct answer from each dropdown menu. consider the following polynomials equations. a = 3x²(x - 1) b = -3x³ + 4x² - 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 • b a polynomial?

Explanation:

Step1: Simplify polynomial A

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

Step2: Compute A + B

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

Step3: Check if A+B is polynomial

This is a polynomial (sum of polynomials is polynomial).

Step4: Compute A - B

$(3x^3 - 3x^2) - (-3x^3 + 4x^2 - 2x + 1) = 3x^3 -3x^2 +3x^3 -4x^2 +2x -1 = 6x^3 -7x^2 +2x -1$

Step5: Check if A-B is polynomial

This is a polynomial (difference of polynomials is polynomial).

Step6: Compute A • B

$(3x^3 - 3x^2)(-3x^3 + 4x^2 - 2x + 1)$
$=3x^3(-3x^3 + 4x^2 - 2x + 1) -3x^2(-3x^3 + 4x^2 - 2x + 1)$
$=-9x^6 +12x^5 -6x^4 +3x^3 +9x^5 -12x^4 +6x^3 -3x^2$
$=-9x^6 +(12x^5+9x^5)+(-6x^4-12x^4)+(3x^3+6x^3)-3x^2$
$=-9x^6 +21x^5 -18x^4 +9x^3 -3x^2$

Step7: Check if A•B is polynomial

This is a polynomial (product of polynomials is polynomial).

Answer:

Is the result of $A + B$ a polynomial? $\boldsymbol{\checkmark}$
Is the result of $A - B$ a polynomial? $\boldsymbol{\checkmark}$
Is the result of $A \bullet B$ a polynomial? $\boldsymbol{\checkmark}$