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question 2
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type the correct answer in each box.
mariana and bethany disagree about whether the set of polynomials is closed under addition, subtraction, and multiplication, as the set of integers is. to examine their two positions, they consider the polynomials ( a = 5x - 1 ) and ( b = 2x ). mariana thinks these polynomials are closed under all three operations. bethany argues that these polynomials are closed under only two operations. complete the following statements, and then determine who is correct.
( a + b = square )
( a - b = square )
( a cdot b = 10x^2 + square )
is correct.

Explanation:

Step1: Calculate \( A + B \)

Given \( A = 5x - 1 \) and \( B = 2x \), we add the two polynomials.
\( A + B=(5x - 1)+(2x) \)
Combine like terms: \( 5x+2x - 1 = 7x - 1 \)

Step2: Calculate \( A - B \)

Subtract \( B \) from \( A \).
\( A - B=(5x - 1)-(2x) \)
Distribute the negative sign: \( 5x - 1 - 2x \)
Combine like terms: \( 3x - 1 \)

Step3: Calculate \( A\cdot B \)

We know \( A\cdot B = 10x^{2}+ \) (let's find the missing term).
\( A\cdot B=(5x - 1)(2x)=10x^{2}-2x \) (using distributive property: \( 5x\times2x=10x^{2} \), \( 5x\times0 = 0 \), \( - 1\times2x=-2x \), \( -1\times0 = 0 \))

Now, to check closure:

• For addition: \( A + B = 7x - 1 \) (a polynomial)
• For subtraction: \( A - B = 3x - 1 \) (a polynomial)
• For multiplication: \( A\cdot B = 10x^{2}-2x \) (a polynomial)

So the set of polynomials is closed under all three operations, so Mariana is correct.

Answer:

\( A + B=\boldsymbol{7x - 1} \)
\( A - B=\boldsymbol{3x - 1} \)
\( A\cdot B = 10x^{2}+\boldsymbol{(-2x)} \) (or \( 10x^{2}-2x \))
Mariana is correct.