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Which of the following correctly simplifies (9x ^ 2 + 17x + 13) - (2x ^ 2 + 12x + 11) ?

12x^2 + 2
7x^2 + 29x + 24
7x^2 + 5x + 2
9x^2 + 17x + 13 - 2x^2 - 12x - 11

37 answers

To simplify the expression \((9x^2 + 17x + 13) - (2x^2 + 12x + 11)\), we start by distributing the negative sign across the second polynomial:

\[ (9x^2 + 17x + 13) - (2x^2 + 12x + 11) = 9x^2 + 17x + 13 - 2x^2 - 12x - 11 \]

Now, we combine like terms:

  1. Combine the \(x^2\) terms: \[ 9x^2 - 2x^2 = 7x^2 \]

  2. Combine the \(x\) terms: \[ 17x - 12x = 5x \]

  3. Combine the constant terms: \[ 13 - 11 = 2 \]

Putting it all together, we have:

\[ 7x^2 + 5x + 2 \]

So, the correctly simplified expression is:

\[ \boxed{7x^2 + 5x + 2} \]

Which of the following correctly simplifies (9x ^ 2 + 17x + 13) - (2x ^ 2 + 12x + 11) ?

12x^2 + 2
7x^2 + 29x + 24
7x^2 + 5x + 2
9x^2 + 17x + 13 - 2x^2 - 12x - 11

To simplify the expression \((9x^2 + 17x + 13) - (2x^2 + 12x + 11)\), we first distribute the negative sign:

\[ (9x^2 + 17x + 13) - (2x^2 + 12x + 11) = 9x^2 + 17x + 13 - 2x^2 - 12x - 11 \]

Next, we combine like terms:

  1. For the \(x^2\) terms: \[ 9x^2 - 2x^2 = 7x^2 \]

  2. For the \(x\) terms: \[ 17x - 12x = 5x \]

  3. For the constant terms: \[ 13 - 11 = 2 \]

Putting it all together, we get:

\[ 7x^2 + 5x + 2 \]

Now, let's compare it with the options provided:

  1. 12x^2 + 2 (Incorrect)
  2. 7x^2 + 29x + 24 (Incorrect)
  3. 7x^2 + 5x + 2 (Correct)
  4. 9x^2 + 17x + 13 - 2x^2 - 12x - 11 (This is just the original expression in another form, so not simplified)

The correct answer is:

\[ \boxed{7x^2 + 5x + 2} \]

Multiply the polynomials (3x-1) (2x+4).
?x² + ?x + ?

To multiply the polynomials \((3x - 1)(2x + 4)\), we can use the distributive property (also known as FOIL for binomials):

  1. First, multiply the first terms: \[ 3x \cdot 2x = 6x^2 \]

  2. Next, multiply the outer terms: \[ 3x \cdot 4 = 12x \]

  3. Then, multiply the inner terms: \[ -1 \cdot 2x = -2x \]

  4. Finally, multiply the last terms: \[ -1 \cdot 4 = -4 \]

Now, combine all these results together: \[ 6x^2 + 12x - 2x - 4 \]

Combine the like terms (the \(x\) terms): \[ 6x^2 + (12x - 2x) - 4 = 6x^2 + 10x - 4 \]

So the final result is: \[ 6x^2 + 10x - 4 \]

Thus, the polynomials multiply to give:

\[ \boxed{6x^2 + 10x - 4} \]

?x² + ?x + ?

After multiplying the polynomials \((3x - 1)(2x + 4)\), we found the result to be:

\[ 6x^2 + 10x - 4 \]

In the format you requested, it can be represented as:

  • Coefficient of \(x^2\): 6
  • Coefficient of \(x\): 10
  • Constant term: -4

Thus, filling in the blanks for \(?x^2 + ?x + ?\) gives us:

\[ 6x^2 + 10x - 4 \]

In summary:

  • ? = 6 (for \(x^2\))
  • ? = 10 (for \(x\))
  • ? = -4 (for the constant term)

If you'd like it explicitly: \[ 6x^2 + 10x - 4 \]

6x² + 10x + ?

In the expression \(6x^2 + 10x - 4\), the question mark represents the constant term.

Thus, for \(6x^2 + 10x + ?\):

  • The answer is -4.

So, it can be completed as:

\[ 6x^2 + 10x - 4 \]

Which of the following responses shows that polynomials form a closed system under multiplication?

3x^2/4 - 1
1/4x^3 * 5
(x) (1/2 * x)
(sqrt(x))(6x * - 3)

To determine which of the given responses shows that polynomials form a closed system under multiplication, we need to verify if multiplying the given expressions yields another polynomial.

Let's analyze each option:

  1. \( \frac{3x^2}{4} - 1 \):

    • This expression itself is a polynomial (specifically, it's a quadratic polynomial). However, it does not demonstrate multiplication.

  2. \( \frac{1}{4}x^3 * 5 \):

    • This simplifies to \( \frac{5}{4}x^3 \). This is still a polynomial (a cubic polynomial), and it shows multiplication of a polynomial by a constant. Therefore, it demonstrates closure under multiplication.

  3. \( (x)(\frac{1}{2} x) \):

    • This multiplies to \( \frac{1}{2}x^2 \), which is a polynomial (a quadratic polynomial). This shows multiplication of polynomials resulting in another polynomial, hence demonstrating closure.

  4. \( (\sqrt{x})(6x * -3) \):

    • The term \(6x * -3\) simplifies to \(-18x\), and multiplying this by \(\sqrt{x}\) gives \(-18x^{\frac{3}{2}}\). The expression \(-18x^{\frac{3}{2}}\) is not a polynomial because polynomials only have non-negative integer exponents.

Based on this analysis, the options that demonstrate polynomials are closed under multiplication are:

  • \( \frac{1}{4} x^3 * 5 \)
  • \( (x)(\frac{1}{2} x) \)

Thus, both \( \frac{1}{4}x^3 * 5 \) and \( (x)(\frac{1}{2} x) \) show that polynomials form a closed system under multiplication. If you need just one response, either of those would suffice; however, the clearest example showing closure is:

\[ \frac{1}{4}x^3 * 5 \]

This option shows multiplication leading to another polynomial directly.

Which of the following responses shows that polynomials form a closed system under multiplication?
3x^2/4 - 1
1/4x^3 * 5
(x) (1/2 * x)
(sqrt(x))(6x * - 3)

To determine which of the responses shows that polynomials form a closed system under multiplication, we need to check whether each option results in a polynomial when multiplied.

  1. \( \frac{3x^2}{4} - 1 \):

    • This expression is a polynomial (specifically, a quadratic polynomial), but it does not demonstrate multiplication.

  2. \( \frac{1}{4}x^3 * 5 \):

    • This simplifies to \( \frac{5}{4}x^3 \). This result is still a polynomial (a cubic polynomial), demonstrating that the product of a polynomial and a constant is a polynomial. Thus, this shows closure.

  3. \( (x)(\frac{1}{2} x) \):

    • This multiplies to \( \frac{1}{2}x^2 \), which is a polynomial (a quadratic polynomial). This also shows multiplication of polynomials resulting in another polynomial, hence demonstrating closure.

  4. \( (\sqrt{x})(6x * -3) \):

    • The term \(6x * -3\) simplifies to \(-18x\), and multiplying this by \(\sqrt{x}\) gives \(-18x^{\frac{3}{2}}\). The expression \(-18x^{\frac{3}{2}}\) is not a polynomial because polynomials only have non-negative integer exponents.

In conclusion:

  • The responses that demonstrate that polynomials form a closed system under multiplication are:

  • \( \frac{1}{4}x^3 * 5 \)

  • \( (x)(\frac{1}{2}x) \)

Both show that multiplying polynomials results in another polynomial.

If one response must be chosen to best demonstrate the closure property of polynomials under multiplication, I would select:

\[ \frac{1}{4}x^3 * 5 \]

This option explicitly illustrates the multiplication of a polynomial by a constant, yielding another polynomial.

Which of the following correctly divides 18x^2 + 9x + 3 by - 3x

6x + 3 + 1/x
-6x - 9 - 3/x
-15x - 3
-6x - 3 - 1/x

To divide the polynomial \(18x^2 + 9x + 3\) by \(-3x\), we can carry out the division term by term.

Step 1: Divide each term

  1. First term: \[ \frac{18x^2}{-3x} = -6x \]

  2. Second term: \[ \frac{9x}{-3x} = -3 \]

  3. Third term: \[ \frac{3}{-3x} = -\frac{1}{x} \]

Step 2: Combine the results

Putting these together, we get: \[ -6x - 3 - \frac{1}{x} \]

Conclusion

So the correct division of \(18x^2 + 9x + 3\) by \(-3x\) is:

\[ \boxed{-6x - 3 - \frac{1}{x}} \]

This matches with the option \(-6x - 3 - \frac{1}{x}\).

Which of the following correctly divides 18x^2 + 9x + 3 by - 3x
6x + 3 + 1/x
-6x - 9 - 3/x
-15x - 3
-6x - 3 - 1/x

To divide the polynomial \(18x^2 + 9x + 3\) by \(-3x\), we can perform the division for each term in the polynomial.

Step 1: Divide each term by \(-3x\)

  1. First term: \[ \frac{18x^2}{-3x} = -6x \]

  2. Second term: \[ \frac{9x}{-3x} = -3 \]

  3. Third term: \[ \frac{3}{-3x} = -\frac{1}{x} \]

Step 2: Combine the results

Putting all the results together, we get: \[ -6x - 3 - \frac{1}{x} \]

Conclusion

The correct answer is: \[ \boxed{-6x - 3 - \frac{1}{x}} \]

This matches with the option \(-6x - 3 - \frac{1}{x}\).

The design for a rectangular solid shipping container has a volume of 24x^3 + 6x^2 - 9x where x represents inches. The box has a height of 3x. Find the container's cross-sectional area using polynomial division. (1 point)

cross-sectional area = ?x^? + ?x - ?

To find the cross-sectional area of the rectangular solid shipping container given its volume as \(24x^3 + 6x^2 - 9x\) and the height as \(3x\), we will perform polynomial division to divide the volume by the height.

Step 1: Set up the division

We need to divide \(24x^3 + 6x^2 - 9x\) by \(3x\).

Step 2: Perform the division

  1. Divide the leading term: \[ \frac{24x^3}{3x} = 8x^2 \]

  2. Multiply and subtract:

    • Multiply \(8x^2\) by \(3x\): \[ 8x^2 \cdot 3x = 24x^3 \]
    • Subtract from the original polynomial: \[ (24x^3 + 6x^2 - 9x) - 24x^3 = 6x^2 - 9x \]

  3. Continue dividing:

    • Now, divide the leading term of the new polynomial \(6x^2\) by \(3x\): \[ \frac{6x^2}{3x} = 2x \]
    • Multiply \(2x\) by \(3x\): \[ 2x \cdot 3x = 6x^2 \]
    • Subtract: \[ (6x^2 - 9x) - 6x^2 = -9x \]

  4. Final step:

    • Now divide \(-9x\) by \(3x\): \[ \frac{-9x}{3x} = -3 \]
    • Multiply \(-3\) by \(3x\): \[ -3 \cdot 3x = -9x \]
    • Subtract: \[ -9x - (-9x) = 0 \]

Step 3: Result of the division

The result of the polynomial division is: \[ 8x^2 + 2x - 3 \]

Conclusion

The cross-sectional area of the container is: \[ \boxed{8x^2 + 2x - 3} \]

This can be expressed in the required format:

  • Cross-sectional area = \(8x^2 + 2x - 3\)
The design for a rectangular solid shipping container has a volume of 24x^3 + 6x^2 - 9x where x represents inches. The box has a height of 3x. Find the container's cross-sectional area using polynomial division.
cross-sectional area = ?x^? + ?x - ?

To find the cross-sectional area of the rectangular container, we need to divide the volume of the container by its height. The given volume \(V\) is:

\[ V = 24x^3 + 6x^2 - 9x \]

and the height \(h\) is:

\[ h = 3x. \]

We will perform polynomial division to divide the volume by the height.

Step 1: Set up the division

We will divide \(24x^3 + 6x^2 - 9x\) by \(3x\).

Step 2: Perform polynomial division

  1. Divide the leading term: \[ \frac{24x^3}{3x} = 8x^2. \]

  2. Multiply and subtract:

    • Multiply \(8x^2\) by \(3x\): \[ 8x^2 \cdot 3x = 24x^3. \]
    • Subtract this from the original polynomial: \[ (24x^3 + 6x^2 - 9x) - 24x^3 = 6x^2 - 9x. \]

  3. Continue dividing:

    • Now take the next term \(6x^2\) and divide by \(3x\): \[ \frac{6x^2}{3x} = 2x. \]
    • Multiply \(2x\) by \(3x\): \[ 2x \cdot 3x = 6x^2. \]
    • Subtract this: \[ (6x^2 - 9x) - 6x^2 = -9x. \]

  4. Final step:

    • Now take \(-9x\) and divide by \(3x\): \[ \frac{-9x}{3x} = -3. \]
    • Multiply \(-3\) by \(3x\): \[ -3 \cdot 3x = -9x. \]
    • Subtract: \[ -9x - (-9x) = 0. \]

Step 3: Result of the division

The result of the polynomial division is: \[ 8x^2 + 2x - 3. \]

Conclusion

The cross-sectional area of the container is: \[ \boxed{8x^2 + 2x - 3}, \]

which can be filled into the format you provided as:

  • Cross-sectional area = \(8x^2 + 2x - 3\)

Thus, the answer for your specific format is:

  • cross-sectional area = 8x^2 + 2x - 3.
Using the Commutative Property, prove that these algebraic expressions are equivalent. In 3-4 sentences, explain your result.
Expression 1: y * 3x
Expression 2: 3xy

To prove the equivalence of the expressions \(y \cdot 3x\) and \(3xy\) using the Commutative Property of Multiplication, we can rearrange the factors in the multiplication. The Commutative Property states that the order in which we multiply numbers does not affect the product, meaning \(a \cdot b = b \cdot a\).

Starting with Expression 1, \(y \cdot 3x\), we can apply the Commutative Property to switch the order of multiplication: \[ y \cdot 3x = 3 \cdot y \cdot x = 3xy. \] Thus, both expressions yield the same result, confirming that \(y \cdot 3x\) is indeed equivalent to \(3xy\). This demonstrates that the expressions are interchangeable due to the associativity and commutativity of multiplication.

Use the Associative Property to demonstrate that Expression 1 is equivalent to Expression 2.
Expression 1: 22r - 13
Expression 2: 19r + (3r - 2) - 11

To demonstrate that the expressions \(22r - 13\) and \(19r + (3r - 2) - 11\) are equivalent using the Associative Property, we will manipulate Expression 2.

Step 1: Simplify Expression 2

Starting with Expression 2: \[ 19r + (3r - 2) - 11. \]

Step 2: Apply the Associative Property

To simplify this, we can first handle the parentheses. The Associative Property allows us to regroup terms when adding or subtracting.

  1. Combine the terms inside the parentheses: \[ 19r + 3r - 2 - 11. \]

  2. Combine like terms (the \(r\) terms): \[ (19r + 3r) - 2 - 11 = 22r - 2 - 11. \]

  3. Combine the constant terms: \[ 22r - (2 + 11) = 22r - 13. \]

Step 3: Conclusion

Having simplified Expression 2 to \(22r - 13\), we see that:

\[ 22r - 13 = 22r - 13, \]

which shows that Expression 1 and Expression 2 are indeed equivalent. This demonstration illustrates the Associative Property, which allows us to regroup and rearrange the terms without changing the result. Thus, both expressions are equal.

Use the Distributive Property to prove that these algebraic expressions are equivalent. Explain your answer in one sentence for each expression and show your work.
Expression 1: (x - 5) * (x + 3)
Expression 2: 2x^2 - x^2 - 2x - 20 + 5

To prove that the expressions \( (x - 5)(x + 3) \) and \( 2x^2 - x^2 - 2x - 20 + 5 \) are equivalent using the Distributive Property, we will expand Expression 1 and simplify Expression 2.

Step 1: Expand Expression 1 using the Distributive Property

Using the Distributive Property (FOIL method for binomials) on \( (x - 5)(x + 3) \):

\[ (x - 5)(x + 3) = x \cdot x + x \cdot 3 - 5 \cdot x - 5 \cdot 3 \]

Calculating each term:

  • \( x \cdot x = x^2 \)
  • \( x \cdot 3 = 3x \)
  • \(-5 \cdot x = -5x\)
  • \(-5 \cdot 3 = -15\)

Now combine the terms:

\[ x^2 + 3x - 5x - 15 = x^2 - 2x - 15. \]

Step 2: Simplify Expression 2

Now look at Expression 2:

\[ 2x^2 - x^2 - 2x - 20 + 5. \]

  1. Combine the like terms \( 2x^2 - x^2 \): \[ 2x^2 - x^2 = x^2. \]

  2. Combine constant terms \( -20 + 5 \): \[ -20 + 5 = -15. \]

Putting it all together, we get:

\[ Expression 2 = x^2 - 2x - 15. \]

Step 3: Conclusion

Now, both expressions have simplified to:

Expression 1:
\[ (x - 5)(x + 3) = x^2 - 2x - 15. \]

Expression 2:
\[ 2x^2 - x^2 - 2x - 20 + 5 = x^2 - 2x - 15. \]

This guarantees that both expressions are equivalent: \[ \boxed{x^2 - 2x - 15}. \]

In summary:

  • For Expression 1, we used the Distributive Property to expand \( (x - 5)(x + 3) \) yielding \( x^2 - 2x - 15 \).
  • For Expression 2, we combined like terms in \( 2x^2 - x^2 - 2x - 20 + 5 \) to also yield \( x^2 - 2x - 15 \).

Thus, both expressions are equivalent as shown.

Use the Distributive Property to prove that these algebraic expressions are equivalent. Explain your answer in 2-3 sentences for each expression and show your work.

Expression 1: (x - 5) * (x + 3)
Expression 2: 2x^2 - x^2 - 2x - 20 + 5

To prove that the expressions \( (x - 5)(x + 3) \) and \( 2x^2 - x^2 - 2x - 20 + 5 \) are equivalent using the Distributive Property, we will expand Expression 1 and simplify Expression 2.

Step 1: Expand Expression 1 using the Distributive Property

Using the Distributive Property (specifically the FOIL method for binomials), we can expand \( (x - 5)(x + 3) \):

\[ (x - 5)(x + 3) = x \cdot x + x \cdot 3 - 5 \cdot x - 5 \cdot 3. \]

Calculating each term:

  • The first term: \(x \cdot x = x^2\).
  • The outer term: \(x \cdot 3 = 3x\).
  • The inner term: \(-5 \cdot x = -5x\).
  • The last term: \(-5 \cdot 3 = -15\).

Now, combine the terms:

\[ x^2 + 3x - 5x - 15 = x^2 - 2x - 15. \]

Thus, Expression 1 simplifies to \(x^2 - 2x - 15\).

Step 2: Simplify Expression 2

Next, we will simplify Expression 2:

\[ 2x^2 - x^2 - 2x - 20 + 5. \]

  1. Combine the like terms \(2x^2 - x^2\): \[ 2x^2 - x^2 = x^2. \]

  2. Now, deal with the constants: \(-20 + 5\): \[ -20 + 5 = -15. \]

Putting it all together:

  • The remaining terms are \(x^2 - 2x - 15\).

Therefore, Expression 2 simplifies to \(x^2 - 2x - 15\).

Conclusion

Now both expressions have been simplified to the same form:

Expression 1: \((x - 5)(x + 3) = x^2 - 2x - 15\).

Expression 2: \(2x^2 - x^2 - 2x - 20 + 5 = x^2 - 2x - 15\).

This shows that both expressions are equivalent, as they yield the same result, \(x^2 - 2x - 15\). Thus, we have demonstrated their equivalence using the Distributive Property.

Use the Distributive Property to prove that these algebraic expressions are equivalent. Explain your answer in 2-3 sentences for each expression.

Expression 1: (x - 5) * (x + 3)
Expression 2: 2x^2 - x^2 - 2x - 20 + 5

To demonstrate that the expressions \( (x - 5)(x + 3) \) and \( 2x^2 - x^2 - 2x - 20 + 5 \) are equivalent using the Distributive Property, we will expand Expression 1 and simplify Expression 2.

Expression 1: \( (x - 5)(x + 3) \)

We can use the Distributive Property (FOIL method) to expand \( (x - 5)(x + 3) \): \[ (x - 5)(x + 3) = x(x) + x(3) - 5(x) - 5(3). \]

Calculating each term gives us:

  • First: \( x^2 \)
  • Outer: \( 3x \)
  • Inner: \( -5x \)
  • Last: \( -15 \)

Combining these results: \[ x^2 + 3x - 5x - 15 = x^2 - 2x - 15. \]

Thus, after expanding Expression 1, we find it simplifies to \( x^2 - 2x - 15 \).

Expression 2: \( 2x^2 - x^2 - 2x - 20 + 5 \)

Now let’s simplify Expression 2 step by step:

  1. Start with \( 2x^2 - x^2 \): \[ 2x^2 - x^2 = x^2. \]

  2. Then, looking at the constants, we combine \(-20 + 5\): \[ -20 + 5 = -15. \]

Putting it all together, we obtain: \[ x^2 - 2x - 15. \]

Thus, Expression 2 simplifies to \( x^2 - 2x - 15 \).

Conclusion

Both expressions are now in the form \( x^2 - 2x - 15 \), confirming their equivalence. Expression 1 simplifies to \( x^2 - 2x - 15 \) using the Distributive Property, while Expression 2 also simplifies to the same result through combining like terms. Therefore, we have shown that these two expressions are indeed equivalent.

Use the Distributive Property to prove that these algebraic expressions are equivalent. Explain your answer in 2-3 sentences for each expression.

Expression 1: (x - 5) * (x + 3)
Expression 2: 2x^2 - x^2 - 2x - 20 + 5
(original and like a 9th grader wrote it)

To show that the expressions \( (x - 5)(x + 3) \) and \( 2x^2 - x^2 - 2x - 20 + 5 \) are equivalent, we will use the Distributive Property, which allows us to multiply terms and combine like terms.

Expression 1: \( (x - 5)(x + 3) \)

Using the Distributive Property (also known as the FOIL method for binomials), we expand the expression: \[ (x - 5)(x + 3) = x \cdot x + x \cdot 3 - 5 \cdot x - 5 \cdot 3. \]

Calculating each part:

  • First: \( x \cdot x = x^2 \).
  • Outer: \( x \cdot 3 = 3x \).
  • Inner: \( -5 \cdot x = -5x \).
  • Last: \( -5 \cdot 3 = -15 \).

Now we combine these results: \[ x^2 + 3x - 5x - 15 = x^2 - 2x - 15. \]

So, Expression 1 simplifies to \( x^2 - 2x - 15 \).

Expression 2: \( 2x^2 - x^2 - 2x - 20 + 5 \)

Now, let's simplify Expression 2 step by step:

  1. Start with \( 2x^2 - x^2 \): \[ 2x^2 - x^2 = x^2. \]

  2. Next, combine the constants \(-20 + 5\): \[ -20 + 5 = -15. \]

Putting everything together, we have: \[ x^2 - 2x - 15. \]

Conclusion

Now both expressions are the same: \( x^2 - 2x - 15 \). Expression 1 was expanded using the Distributive Property (FOIL), and Expression 2 was simplified by combining like terms. Since both simplify to the same result, we have proven that \( (x - 5)(x + 3) \) and \( 2x^2 - x^2 - 2x - 20 + 5 \) are equivalent.

In 3-4 sentences, explain why the following operation is a closed system under subtraction.
(9x^2 + 17x + 13) - (2x^2 + 12x + 11)

(original and like a 9th grader wrote it)

The operation of subtraction is a closed system under polynomials because when we subtract one polynomial from another, the result is still a polynomial. For example, in the expression \((9x^2 + 17x + 13) - (2x^2 + 12x + 11)\), we are subtracting two polynomials. When we perform the subtraction, we combine like terms and get another polynomial, which shows that the set of polynomials remains unchanged. This means that the result of the subtraction fits within the same category of polynomials, proving that it is closed under this operation.

Use the following multiplication problem to show how multiplying polynomials is a closed system. (x^2 - 1)(-2x + 5)

To demonstrate that multiplying polynomials is a closed system, we can take the example of the multiplication \( (x^2 - 1)(-2x + 5) \). When we multiply two polynomials, the result is another polynomial, which shows that the set of polynomials is closed under multiplication.

Step 1: Distribute Each Term

We will use the Distributive Property (also known as the FOIL method for binomials) to solve the multiplication:

  1. First, multiply \(x^2\) by each term in \((-2x + 5)\):

    • \(x^2 \cdot (-2x) = -2x^3\)
    • \(x^2 \cdot 5 = 5x^2\)

  2. Next, multiply \(-1\) by each term in \((-2x + 5)\):

    • \(-1 \cdot (-2x) = 2x\)
    • \(-1 \cdot 5 = -5\)

Step 2: Combine All the Results

Now, let’s put all these results together: \[ -2x^3 + 5x^2 + 2x - 5. \]

Step 3: Conclusion

The final expression, \(-2x^3 + 5x^2 + 2x - 5\), is a polynomial. Since we started with two polynomials and ended up with a polynomial, this illustrates that multiplying polynomials is a closed system. No matter how many polynomials we multiply or what polynomials we choose, the result will always be another polynomial, proving that the operation is closed within the set of polynomials.

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