Asked by x
Q15 Using the Commutative Property, prove that these algebraic expressions are equivalent. In 1–2 sentences explain your result.
Expression 1: z ⋅ 4x
Expression 2: 4xz
Expression 1: z ⋅ 4x
Expression 2: 4xz
Answers
Answered by
x
Q16 Use the Associative Property to demonstrate that Expression 1 is equivalent to Expression 2.
Expression 1:25r - 15
Expression 2: 20r + (5r - 3) - 12
Expression 1:25r - 15
Expression 2: 20r + (5r - 3) - 12
Answered by
x
Q17 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-7)(x+4)
Expression 2: x2- 3x - 28
Expression 1:(x-7)(x+4)
Expression 2: x2- 3x - 28
Answered by
x
Q17 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-7)(x+4)
Expression 2: x^2- 3x - 28
Expression 1:(x-7)(x+4)
Expression 2: x^2- 3x - 28
Answered by
x
Q18 In 1–2 sentences explain why the following operation is a closed system under subtraction.
(8x^2+14x+10)-(3x^2+9x+8)
(8x^2+14x+10)-(3x^2+9x+8)
Answered by
x
Q19 Use the following multiplication problem to show how multiplying polynomials is a closed system.
(x^2 - 2)(-3x + 7)
(x^2 - 2)(-3x + 7)
Answered by
GPT-4o mini
To prove that the expressions \( z \cdot 4x \) and \( 4xz \) are equivalent using the Commutative Property of multiplication, we can rearrange the factors in Expression 1:
\[
z \cdot 4x = 4 \cdot z \cdot x = 4xz.
\]
Thus, both expressions simplify to the same result, demonstrating their equivalence. The Commutative Property allows us to change the order of multiplication without affecting the outcome, confirming that \( z \cdot 4x \) is equal to \( 4xz \).
\[
z \cdot 4x = 4 \cdot z \cdot x = 4xz.
\]
Thus, both expressions simplify to the same result, demonstrating their equivalence. The Commutative Property allows us to change the order of multiplication without affecting the outcome, confirming that \( z \cdot 4x \) is equal to \( 4xz \).
Answered by
GPT-4o mini
To demonstrate that Expression 1, \( 25r - 15 \), is equivalent to Expression 2, \( 20r + (5r - 3) - 12 \), we will simplify Expression 2 using the Associative Property.
Starting with Expression 2:
\[
20r + (5r - 3) - 12
\]
First, we can simplify the parentheses:
\[
20r + 5r - 3 - 12
\]
Next, we can combine the constants:
\[
20r + 5r - 15
\]
Now, combining the like terms \(20r\) and \(5r\):
\[
(20r + 5r) - 15 = 25r - 15
\]
This shows that:
\[
25r - 15 = 25r - 15
\]
Thus, we have demonstrated that \( 25r - 15 \) and \( 20r + (5r - 3) - 12 \) are equivalent using the Associative Property. The rearrangement of terms and grouping allowed us to see that both expressions yield the same result.
Starting with Expression 2:
\[
20r + (5r - 3) - 12
\]
First, we can simplify the parentheses:
\[
20r + 5r - 3 - 12
\]
Next, we can combine the constants:
\[
20r + 5r - 15
\]
Now, combining the like terms \(20r\) and \(5r\):
\[
(20r + 5r) - 15 = 25r - 15
\]
This shows that:
\[
25r - 15 = 25r - 15
\]
Thus, we have demonstrated that \( 25r - 15 \) and \( 20r + (5r - 3) - 12 \) are equivalent using the Associative Property. The rearrangement of terms and grouping allowed us to see that both expressions yield the same result.
Answered by
GPT-4o mini
To prove that Expression 1, \( (x - 7)(x + 4) \), is equivalent to Expression 2, \( x^2 - 3x - 28 \), we can use the Distributive Property (also known as the FOIL method for binomials) to expand Expression 1.
**Step 1: Expand Expression 1 using the Distributive Property.**
\[
(x - 7)(x + 4) = x \cdot x + x \cdot 4 - 7 \cdot x - 7 \cdot 4
\]
Calculating each part:
\[
= x^2 + 4x - 7x - 28
\]
Combining the like terms \(4x\) and \(-7x\):
\[
= x^2 - 3x - 28
\]
**Conclusion:**
Thus, Expression 1 simplifies to \( x^2 - 3x - 28 \), which matches Expression 2. This shows that both expressions are equivalent by fully expanding Expression 1 using the Distributive Property.
**Step 1: Expand Expression 1 using the Distributive Property.**
\[
(x - 7)(x + 4) = x \cdot x + x \cdot 4 - 7 \cdot x - 7 \cdot 4
\]
Calculating each part:
\[
= x^2 + 4x - 7x - 28
\]
Combining the like terms \(4x\) and \(-7x\):
\[
= x^2 - 3x - 28
\]
**Conclusion:**
Thus, Expression 1 simplifies to \( x^2 - 3x - 28 \), which matches Expression 2. This shows that both expressions are equivalent by fully expanding Expression 1 using the Distributive Property.
Answered by
GPT-4o mini
To prove that Expression 1, \( (x - 7)(x + 4) \), is equivalent to Expression 2, \( x^2 - 3x - 28 \), we can use the Distributive Property (often referred to as the FOIL method for multiplying binomials).
### Step 1: Expand Expression 1 using the Distributive Property.
\[
(x - 7)(x + 4)
\]
Applying the Distributive Property (FOIL method):
- **First:** \( x \cdot x = x^2 \)
- **Outside:** \( x \cdot 4 = 4x \)
- **Inside:** \( -7 \cdot x = -7x \)
- **Last:** \( -7 \cdot 4 = -28 \)
Now combine these results:
\[
x^2 + 4x - 7x - 28
\]
### Step 2: Combine like terms.
\[
x^2 + (4x - 7x) - 28
\]
\[
= x^2 - 3x - 28
\]
### Conclusion:
Thus, we have shown that Expression 1 simplifies to \( x^2 - 3x - 28 \), which is exactly Expression 2. This confirms that both expressions are equivalent by expanding Expression 1 using the Distributive Property.
### Step 1: Expand Expression 1 using the Distributive Property.
\[
(x - 7)(x + 4)
\]
Applying the Distributive Property (FOIL method):
- **First:** \( x \cdot x = x^2 \)
- **Outside:** \( x \cdot 4 = 4x \)
- **Inside:** \( -7 \cdot x = -7x \)
- **Last:** \( -7 \cdot 4 = -28 \)
Now combine these results:
\[
x^2 + 4x - 7x - 28
\]
### Step 2: Combine like terms.
\[
x^2 + (4x - 7x) - 28
\]
\[
= x^2 - 3x - 28
\]
### Conclusion:
Thus, we have shown that Expression 1 simplifies to \( x^2 - 3x - 28 \), which is exactly Expression 2. This confirms that both expressions are equivalent by expanding Expression 1 using the Distributive Property.
Answered by
GPT-4o mini
The operation \( (8x^2 + 14x + 10) - (3x^2 + 9x + 8) \) is a closed system under subtraction because subtracting two polynomials results in another polynomial. Performing this operation yields \( (8x^2 - 3x^2) + (14x - 9x) + (10 - 8) = 5x^2 + 5x + 2 \), which is still a polynomial expression.
Answered by
GPT-4o mini
To show that multiplying polynomials is a closed system, we will expand the expression \( (x^2 - 2)(-3x + 7) \) using the Distributive Property, which involves distributing each term in the first polynomial by each term in the second polynomial.
### Step 1: Distribute each term in \( (x^2 - 2) \).
1. Multiply \( x^2 \) by \( -3x \):
\[
x^2 \cdot (-3x) = -3x^3
\]
2. Multiply \( x^2 \) by \( 7 \):
\[
x^2 \cdot 7 = 7x^2
\]
3. Multiply \( -2 \) by \( -3x \):
\[
-2 \cdot (-3x) = 6x
\]
4. Multiply \( -2 \) by \( 7 \):
\[
-2 \cdot 7 = -14
\]
### Step 2: Combine all the results.
Now, combine all the terms obtained from the distribution:
\[
-3x^3 + 7x^2 + 6x - 14
\]
### Conclusion:
The resulting expression, \( -3x^3 + 7x^2 + 6x - 14 \), is a polynomial, demonstrating that the product of two polynomials is still a polynomial. This confirms that multiplying polynomials is a closed system, as the result remains within the set of polynomials.
### Step 1: Distribute each term in \( (x^2 - 2) \).
1. Multiply \( x^2 \) by \( -3x \):
\[
x^2 \cdot (-3x) = -3x^3
\]
2. Multiply \( x^2 \) by \( 7 \):
\[
x^2 \cdot 7 = 7x^2
\]
3. Multiply \( -2 \) by \( -3x \):
\[
-2 \cdot (-3x) = 6x
\]
4. Multiply \( -2 \) by \( 7 \):
\[
-2 \cdot 7 = -14
\]
### Step 2: Combine all the results.
Now, combine all the terms obtained from the distribution:
\[
-3x^3 + 7x^2 + 6x - 14
\]
### Conclusion:
The resulting expression, \( -3x^3 + 7x^2 + 6x - 14 \), is a polynomial, demonstrating that the product of two polynomials is still a polynomial. This confirms that multiplying polynomials is a closed system, as the result remains within the set of polynomials.