Johnny rewrites the expression, 2x2−3x−3 ,as the expression shown below. Explain why this statement is or is not equivalent to the original expression. 2x2−(3x+3) (1 point) Responses No, they are NOT equivalent because if you distribute the negative sign, the last number will be positive 3 and in the original expression it is -3. No, they are NOT equivalent because if you distribute the negative sign, the last number will be positive 3 and in the original expression it is -3. No, they are NOT equivalent because if you simplify that expression, you get -6x at the very end and that is not in the original expression. No, they are NOT equivalent because if you simplify that expression, you get -6x at the very end and that is not in the original expression. Yes, they are equivalent because if you simplify the expression, you get a 6x at the end and the original expression has a 6x. Yes, they are equivalent because if you simplify the expression, you get a 6x at the end and the original expression has a 6x. Yes, they are equivalent because if you distribute the negative sign, you get -3x and -3 and that is the same in the simplified original expression .

To determine if Johnny's rewritten expression \( 2x^2 - (3x + 3) \) is equivalent to the original expression \( 2x^2 - 3x - 3 \), let's simplify the rewritten expression step by step.

Starting with the rewritten expression:

\[
2x^2 - (3x + 3)
\]

Distributing the negative sign gives us:

\[
2x^2 - 3x - 3
\]

Now we can see that this matches exactly with the original expression \( 2x^2 - 3x - 3 \).

Thus, the expressions are indeed equivalent.

The correct response would be:

**Yes, they are equivalent because if you distribute the negative sign, you get -3x and -3, which is the same as in the original expression.**

Question Pick the expression for that has the same number of terms as the following expression: 2x2−3x−3 (1 point) Responses 7x2+3 7 x squared plus 3 6x2−9x−8 6 x squared minus 9 x minus 8 4x 4x −4

To find an expression that has the same number of terms as the expression \(2x^2 - 3x - 3\), we first need to identify how many terms the original expression has.

The original expression \(2x^2 - 3x - 3\) has three distinct terms:
1. \(2x^2\) (the quadratic term)
2. \(-3x\) (the linear term)
3. \(-3\) (the constant term)

Now, let’s analyze each of the given response options to see how many terms they each have:

1. **\(7x^2 + 3\)**: This has 2 terms.
2. **\(6x^2 - 9x - 8\)**: This has 3 terms.
3. **\(4x\)**: This has 1 term.
4. **\(-4\)**: This has 1 term.

From the above, the only expression that has the same number of terms (3 terms) as \(2x^2 - 3x - 3\) is:

**\(6x^2 - 9x - 8\)**.

So, that would be the correct choice.

Add the following polynomials: (2x2−3x−3 )+(6x2−9x−8) (1 point) Responses 8x2−12x−11 8 x squared minus 12 x minus 11 −4x2+6x+5 negative 4 x squared plus 6 x plus 5 8x2+6x−5 8 x squared plus 6 x minus 5 −4x2−12x+11 negative 4 x squared minus 12 x plus 11

To add the polynomials \( (2x^2 - 3x - 3) + (6x^2 - 9x - 8) \), we can combine like terms.

Let's rewrite the expression for clarity:

\[
(2x^2 - 3x - 3) + (6x^2 - 9x - 8)
\]

Now we'll combine like terms:

1. **Quadratic terms** (\(x^2\)):
- \(2x^2 + 6x^2 = 8x^2\)

2. **Linear terms** (\(x\)):
- \(-3x - 9x = -12x\)

3. **Constant terms**:
- \(-3 - 8 = -11\)

Putting it all together, we have:

\[
8x^2 - 12x - 11
\]

Now let's match this with the provided options:

1. **\(8x^2 - 12x - 11\)**
2. **\(-4x^2 + 6x + 5\)**
3. **\(8x^2 + 6x - 5\)**
4. **\(-4x^2 - 12x + 11\)**

The correct response is:

**\(8x^2 - 12x - 11\)**.

Find the difference: (2x2−3x−3 )−(6x2−9x−8) (1 point) Responses 8x2−12x−11 8 x squared minus 12 x minus 11 −4x2+6x+5 negative 4 x squared plus 6 x plus 5 8x2+6x−5 8 x squared plus 6 x minus 5 −4x2−12x+11

To find the difference \( (2x^2 - 3x - 3) - (6x^2 - 9x - 8) \), we can rewrite the expression for clarity:

\[
(2x^2 - 3x - 3) - (6x^2 - 9x - 8)
\]

This can be simplified by distributing the negative sign across the second polynomial:

\[
= 2x^2 - 3x - 3 - 6x^2 + 9x + 8
\]

Now, we can combine like terms:

1. **Quadratic terms** (\(x^2\)):
- \(2x^2 - 6x^2 = -4x^2\)

2. **Linear terms** (\(x\)):
- \(-3x + 9x = 6x\)

3. **Constant terms**:
- \(-3 + 8 = 5\)

Putting it all together, we get:

\[
-4x^2 + 6x + 5
\]

Now let's match this with the provided options:

1. **\(8x^2 - 12x - 11\)**
2. **\(-4x^2 + 6x + 5\)**
3. **\(8x^2 + 6x - 5\)**
4. **\(-4x^2 - 12x + 11\)**

The correct response is:

**\(-4x^2 + 6x + 5\)**.

The result of adding 2 polynomials results in another polynomial, meaning that polynomials are closed under addition.(1 point) Responses True True False

The statement is **True**.

Polynomials are indeed closed under addition, meaning that when you add two polynomials together, the result is always another polynomial. This is a fundamental property of polynomials.

Johnny wonders if he can relate the sum and difference of two polynomials, using the equations shown. -(K + F) = K - F. Are the two expressions shown equivalent to each other? What evidence supports or disproves the statement? Pick the statement below that clearly explains your answer, remembering to justify by listing the property used. (1 point) Responses YES - the two expressions are equivalent because -(K + F) = K - F and I used the negative property of multiplication to solve. YES - the two expressions are equivalent because -(K + F) = K - F and I used the negative property of multiplication to solve. NO - they are not equivalent because -(K + F) = -K + -F. I used the Distributive Property to simplify the left side. NO - they are not equivalent because -(K + F) = -K + -F. I used the Distributive Property to simplify the left side. YES - they are equivalent because -(K + F) = -K + F and then you can switch the - to the F. I used the Commutative Property to change the signs of the K and the F. YES - they are equivalent because -(K + F) = -K + F and then you can switch the - to the F. I used the Commutative Property to change the signs of the K and the F. NO - they are not equivalent because -(K + F) = -K + F and that is not the correct way it is written above. I used the Distributive Property.

To determine whether the expressions \(- (K + F)\) and \(K - F\) are equivalent, let's analyze the expression \(- (K + F)\) using the Distributive Property.

Starting with:

\[
-(K + F)
\]

Using the Distributive Property, we distribute the negative sign:

\[
-(K + F) = -K - F
\]

Now, we compare this result \(-K - F\) to the expression \(K - F\).

Clearly, \(-K - F\) is not equivalent to \(K - F\).

So, the correct response is:

**NO - they are not equivalent because \(-(K + F) = -K - F\). I used the Distributive Property to simplify the left side.**