Asked by x
Which of the following correctly divides 18x^2+9x+3
by −3x
?(1 point)
Responses
−6x−9−3/x
negative 6 x minus 9 minus Start Fraction 3 over x End Fraction
−6x−3−1/x
negative 6 x minus 3 minus Start Fraction 1 over x End Fraction
6x+3+1/x
6 x plus 3 plus Start Fraction 1 over x End Fraction
−15x−3
by −3x
?(1 point)
Responses
−6x−9−3/x
negative 6 x minus 9 minus Start Fraction 3 over x End Fraction
−6x−3−1/x
negative 6 x minus 3 minus Start Fraction 1 over x End Fraction
6x+3+1/x
6 x plus 3 plus Start Fraction 1 over x End Fraction
−15x−3
Answers
Answered by
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 - ( )
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 - ( )
Answered by
x
Using the Commutative Property, prove that these algebraic expressions are equivalent. In 1–2 sentences, explain your result.
Expression 1: y⋅3x
Expression 2: 3xy
Expression 1: y⋅3x
Expression 2: 3xy
Answered by
x
Use the Associative Property to demonstrate that Expression 1 is equivalent to Expression 2.
Expression 1: 22r−13
Expression 2: 19r+(3r−2)−11
(short explanation)
Expression 1: 22r−13
Expression 2: 19r+(3r−2)−11
(short explanation)
Answered by
x
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
Expression 1: (x−5)(x+3)
Expression 2: 2x^2−x^2−2x−20+5
Answered by
x
In 1–2 sentences, explain why the following operation is a closed system under subtraction.
(9x^2+17x+13)−(2x^2+12x+11)
(9x^2+17x+13)−(2x^2+12x+11)
Answered by
GPT-4o mini
To divide the polynomial \(18x^2 + 9x + 3\) by \(-3x\), we will divide each term of the polynomial by \(-3x\).
1. **Divide \(18x^2\) by \(-3x\)**:
\[
\frac{18x^2}{-3x} = -6x
\]
2. **Divide \(9x\) by \(-3x\)**:
\[
\frac{9x}{-3x} = -3
\]
3. **Divide \(3\) by \(-3x\)**:
\[
\frac{3}{-3x} = -\frac{1}{x}
\]
Now we can combine these results:
\[
-6x - 3 - \frac{1}{x}
\]
Thus, the correct answer is:
\(-6x - 3 - \frac{1}{x}\)
So, the correct response is:
**−6x−3−1/x**
1. **Divide \(18x^2\) by \(-3x\)**:
\[
\frac{18x^2}{-3x} = -6x
\]
2. **Divide \(9x\) by \(-3x\)**:
\[
\frac{9x}{-3x} = -3
\]
3. **Divide \(3\) by \(-3x\)**:
\[
\frac{3}{-3x} = -\frac{1}{x}
\]
Now we can combine these results:
\[
-6x - 3 - \frac{1}{x}
\]
Thus, the correct answer is:
\(-6x - 3 - \frac{1}{x}\)
So, the correct response is:
**−6x−3−1/x**
Answered by
GPT-4o mini
To find the container's cross-sectional area, we need to perform polynomial division of the volume \(24x^3 + 6x^2 - 9x\) by the height \(3x\).
### Step 1: Set up the division
We want to divide:
\[
\frac{24x^3 + 6x^2 - 9x}{3x}
\]
### Step 2: Perform the division term by term
1. **Divide the first term:**
\[
\frac{24x^3}{3x} = 8x^2
\]
2. **Multiply and subtract:**
\[
8x^2 \cdot 3x = 24x^3
\]
\[
(24x^3 + 6x^2 - 9x) - 24x^3 = 6x^2 - 9x
\]
3. **Divide the next term:**
\[
\frac{6x^2}{3x} = 2x
\]
4. **Multiply and subtract:**
\[
2x \cdot 3x = 6x^2
\]
\[
(6x^2 - 9x) - 6x^2 = -9x
\]
5. **Divide the next term:**
\[
\frac{-9x}{3x} = -3
\]
6. **Multiply and subtract:**
\[
-3 \cdot 3x = -9x
\]
\[
(-9x) - (-9x) = 0
\]
### Step 3: Combine the results
After performing the division, we find:
\[
\frac{24x^3 + 6x^2 - 9x}{3x} = 8x^2 + 2x - 3
\]
### Conclusion
The container's cross-sectional area is given by:
\[
\text{cross-sectional area} = 8x^2 + 2x - 3
\]
Thus, in the form provided,
\(\text{cross-sectional area} = (8)^{(2)} + (2)x - (3)\).
### Step 1: Set up the division
We want to divide:
\[
\frac{24x^3 + 6x^2 - 9x}{3x}
\]
### Step 2: Perform the division term by term
1. **Divide the first term:**
\[
\frac{24x^3}{3x} = 8x^2
\]
2. **Multiply and subtract:**
\[
8x^2 \cdot 3x = 24x^3
\]
\[
(24x^3 + 6x^2 - 9x) - 24x^3 = 6x^2 - 9x
\]
3. **Divide the next term:**
\[
\frac{6x^2}{3x} = 2x
\]
4. **Multiply and subtract:**
\[
2x \cdot 3x = 6x^2
\]
\[
(6x^2 - 9x) - 6x^2 = -9x
\]
5. **Divide the next term:**
\[
\frac{-9x}{3x} = -3
\]
6. **Multiply and subtract:**
\[
-3 \cdot 3x = -9x
\]
\[
(-9x) - (-9x) = 0
\]
### Step 3: Combine the results
After performing the division, we find:
\[
\frac{24x^3 + 6x^2 - 9x}{3x} = 8x^2 + 2x - 3
\]
### Conclusion
The container's cross-sectional area is given by:
\[
\text{cross-sectional area} = 8x^2 + 2x - 3
\]
Thus, in the form provided,
\(\text{cross-sectional area} = (8)^{(2)} + (2)x - (3)\).
Answered by
GPT-4o mini
To prove that the expressions \(y \cdot 3x\) and \(3xy\) are equivalent using the Commutative Property, we can rearrange the factors in the first expression:
\[
y \cdot 3x = 3 \cdot y \cdot x = 3xy.
\]
Since multiplication is commutative, the order of the factors does not affect the product. This shows that both expressions are indeed equivalent. Thus, we conclude that \(y \cdot 3x\) and \(3xy\) represent the same product, demonstrating the Commutative Property of multiplication.
\[
y \cdot 3x = 3 \cdot y \cdot x = 3xy.
\]
Since multiplication is commutative, the order of the factors does not affect the product. This shows that both expressions are indeed equivalent. Thus, we conclude that \(y \cdot 3x\) and \(3xy\) represent the same product, demonstrating the Commutative Property of multiplication.
Answered by
GPT-4o mini
To demonstrate that the expressions \(22r - 13\) and \(19r + (3r - 2) - 11\) are equivalent using the Associative Property, we will simplify Expression 2 step by step:
1. Start with Expression 2:
\[
19r + (3r - 2) - 11.
\]
2. Apply the Associative Property to group terms:
\[
19r + 3r - 2 - 11.
\]
3. Combine like terms:
\[
(19r + 3r) + (-2 - 11) = 22r - 13.
\]
Now we see that both expressions simplify to the same result, confirming their equivalence.
**Short Explanation:** By applying the Associative Property to rearrange and group terms in Expression 2, we can simplify it to \(22r - 13\), which shows that it is equivalent to Expression 1.
1. Start with Expression 2:
\[
19r + (3r - 2) - 11.
\]
2. Apply the Associative Property to group terms:
\[
19r + 3r - 2 - 11.
\]
3. Combine like terms:
\[
(19r + 3r) + (-2 - 11) = 22r - 13.
\]
Now we see that both expressions simplify to the same result, confirming their equivalence.
**Short Explanation:** By applying the Associative Property to rearrange and group terms in Expression 2, we can simplify it to \(22r - 13\), which shows that it is equivalent to Expression 1.
Answered by
GPT-4o mini
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 first expand Expression 1 and then simplify Expression 2.
### Expression 1: \((x - 5)(x + 3)\)
Using the Distributive Property (FOIL method) to expand Expression 1:
1. **First**: \(x \cdot x = x^2\)
2. **Outer**: \(x \cdot 3 = 3x\)
3. **Inner**: \(-5 \cdot x = -5x\)
4. **Last**: \(-5 \cdot 3 = -15\)
Now combine these results:
\[
(x - 5)(x + 3) = x^2 + 3x - 5x - 15 = x^2 - 2x - 15
\]
### Expression 2: \(2x^2 - x^2 - 2x - 20 + 5\)
Now, let's simplify Expression 2 by combining like terms:
1. Combine \(2x^2 - x^2\):
\[
2x^2 - x^2 = x^2
\]
2. Now, let's combine other terms:
\[
x^2 - 2x - 20 + 5 = x^2 - 2x - 15
\]
### Conclusion
Both expressions simplify to the same result:
\[
x^2 - 2x - 15.
\]
Thus, \((x - 5)(x + 3) = 2x^2 - x^2 - 2x - 20 + 5\).
**Explanation**:
Using the Distributive Property, Expression 1 expands to \(x^2 - 2x - 15\), which is identical to the simplified form of Expression 2, confirming their equivalence.
### Expression 1: \((x - 5)(x + 3)\)
Using the Distributive Property (FOIL method) to expand Expression 1:
1. **First**: \(x \cdot x = x^2\)
2. **Outer**: \(x \cdot 3 = 3x\)
3. **Inner**: \(-5 \cdot x = -5x\)
4. **Last**: \(-5 \cdot 3 = -15\)
Now combine these results:
\[
(x - 5)(x + 3) = x^2 + 3x - 5x - 15 = x^2 - 2x - 15
\]
### Expression 2: \(2x^2 - x^2 - 2x - 20 + 5\)
Now, let's simplify Expression 2 by combining like terms:
1. Combine \(2x^2 - x^2\):
\[
2x^2 - x^2 = x^2
\]
2. Now, let's combine other terms:
\[
x^2 - 2x - 20 + 5 = x^2 - 2x - 15
\]
### Conclusion
Both expressions simplify to the same result:
\[
x^2 - 2x - 15.
\]
Thus, \((x - 5)(x + 3) = 2x^2 - x^2 - 2x - 20 + 5\).
**Explanation**:
Using the Distributive Property, Expression 1 expands to \(x^2 - 2x - 15\), which is identical to the simplified form of Expression 2, confirming their equivalence.
Answered by
GPT-4o mini
A closed system under subtraction means that when you subtract one element from another within the set, the result remains within the same set. In this case, subtracting \((2x^2 + 12x + 11)\) from \((9x^2 + 17x + 13)\) results in a polynomial expression \((9x^2 + 17x + 13) - (2x^2 + 12x + 11) = 7x^2 + 5x + 2\), which is also a polynomial, thus demonstrating that the operation is closed within the set of polynomials.