select the correct answer from each drop-down menu.

consider the following polynomials equations.
\
\begin{aligned}
a &= 3x^2(x - 1) \\
b &= -3x^3 + 4x^2 - 2x + 1
\end{aligned}
\

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

is the result of $a + b$ a polynomial? choose an answer
is the result of $a - b$ a polynomial? choose an answer
is the result of $a \cdot b$ a polynomial? choose an answer

Question

select the correct answer from each drop-down menu.

consider the following polynomials equations.
\
\begin{aligned}
a &= 3x^2(x - 1) \\
b &= -3x^3 + 4x^2 - 2x + 1
\end{aligned}
\

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

is the result of $a + b$ a polynomial? choose an answer
is the result of $a - b$ a polynomial? choose an answer
is the result of $a \cdot b$ a polynomial? choose an answer

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Polynomial Operations" ], "new_concepts": [ "Polynomial Closure" ], "current_concepts": [ "Polynomial Operations", "Polynomial Closure" ] } ## Simplify the given expressions $$ \begin{aligned} A &= 3x^2(x - 1) = 3x^3 - 3x^2 \ B &= -3x^3 + 4x^2 - 2x + 1 \end{aligned} $$ ## Evaluate each operation $$ \begin{aligned} A + B &= (3x^3 - 3x^2) + (-3x^3 + 4x^2 - 2x + 1) = x^2 - 2x + 1 \ A - B &= (3x^3 - 3x^2) - (-3x^3 + 4x^2 - 2x + 1) = 6x^3 - 7x^2 + 2x - 1 \ A \cdot B &= (3x^3 - 3x^2)(-3x^3 + 4x^2 - 2x + 1) = -9x^6 + 21x^5 - 18x^4 + 9x^3 - 3x^2 \end{aligned} $$ ## Determine polynomial status $$ \begin{aligned} &\text{All results contain only non-negative integer exponents of } x.\ &\text{Therefore, } A+B, A-B, \text{ and } A \cdot B \text{ are all polynomials.} \end{aligned} $$ Consider the following polynomials equations. $$ \begin{aligned} A &= 3x^2(x - 1) \ B &= -3x^3 + 4x^2 - 2x + 1 \end{aligned} $$ Perform each operation and determine if the result is a polynomial. Is the result of $A + B$ a polynomial? Yes Is the result of $A - B$ a polynomial? Yes Is the result of $A \cdot B$ a polynomial? Yes { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Polynomial Operations" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Simplify the given expressions

Evaluate each operation

Determine polynomial status

Answer: