Asked by bluesanta
Multiply the polynomial expression (x+2)(x−4)
(needs number)x^2 + (needs number)x +(needs number)
(needs number)x^2 + (needs number)x +(needs number)
Answers
Answered by
bluesanta
Multiply to simplify the polynomial expression (r−1)(r2−2r+3)
.(2 points)
r3+
r2+
r+
.(2 points)
r3+
r2+
r+
Answered by
bluesanta
Simplify the polynomial expression (xy−2)(x2+1)
.(2 points)
x3y+
x2+
xy+
.(2 points)
x3y+
x2+
xy+
Answered by
bluesanta
Which of the following shows that polynomials form a closed system under multiplication?
Option #1: 3(2x+5)=6x+15
Option #2: 8x+7=5x+3x+1+6
Option #3: (2x2−4)(3y+6)=6x2y+12x2−12y−24
Option #4: 2x12⋅3=6x−−√
Option #1: 3(2x+5)=6x+15
Option #2: 8x+7=5x+3x+1+6
Option #3: (2x2−4)(3y+6)=6x2y+12x2−12y−24
Option #4: 2x12⋅3=6x−−√
Answered by
bluesanta
Multiply the polynomial (b+8)(3b−6) to simplify.(1 point)
Responses
3b2+18b−48
3 b squared plus 18 b minus 48
3b2+30b−48
3 b squared plus 30 b minus 48
b2+30b−48
b squared plus 30 b minus 48
b2+18b−36
Responses
3b2+18b−48
3 b squared plus 18 b minus 48
3b2+30b−48
3 b squared plus 30 b minus 48
b2+30b−48
b squared plus 30 b minus 48
b2+18b−36
Answered by
bluesanta
What is the product of the polynomials (x2y+2)(x2−y)?(1 point)
Responses
x3y−x2y2+2x2−2y
x cubed y minus x squared y squared plus 2 x squared minus 2 y
x4y−xy+x2y
x superscript 4 baseline y minus x y plus x squared y
x4y−x2y2+2x2−2y
x superscript 4 baseline y minus x squared y squared plus 2 x squared minus 2 y
x4y+x2−2y
Responses
x3y−x2y2+2x2−2y
x cubed y minus x squared y squared plus 2 x squared minus 2 y
x4y−xy+x2y
x superscript 4 baseline y minus x y plus x squared y
x4y−x2y2+2x2−2y
x superscript 4 baseline y minus x squared y squared plus 2 x squared minus 2 y
x4y+x2−2y
Answered by
GPT-4o mini
To multiply the polynomial expression \((x + 2)(x - 4)\), we can use the distributive property (also known as the FOIL method for binomials). Here are the steps:
1. Multiply the first terms:
\[
x \cdot x = x^2
\]
2. Multiply the outer terms:
\[
x \cdot (-4) = -4x
\]
3. Multiply the inner terms:
\[
2 \cdot x = 2x
\]
4. Multiply the last terms:
\[
2 \cdot (-4) = -8
\]
Now, combine all these results:
\[
x^2 - 4x + 2x - 8
\]
Combine the like terms \(-4x\) and \(2x\):
\[
x^2 - 2x - 8
\]
Thus, the final result is:
\[
x^2 - 2x - 8
\]
So, the expression can be written as:
\[
1x^2 + (-2)x + (-8)
\]
In the format you requested:
- The coefficient of \(x^2\) is **1**.
- The coefficient of \(x\) is **-2**.
- The constant term is **-8**.
1. Multiply the first terms:
\[
x \cdot x = x^2
\]
2. Multiply the outer terms:
\[
x \cdot (-4) = -4x
\]
3. Multiply the inner terms:
\[
2 \cdot x = 2x
\]
4. Multiply the last terms:
\[
2 \cdot (-4) = -8
\]
Now, combine all these results:
\[
x^2 - 4x + 2x - 8
\]
Combine the like terms \(-4x\) and \(2x\):
\[
x^2 - 2x - 8
\]
Thus, the final result is:
\[
x^2 - 2x - 8
\]
So, the expression can be written as:
\[
1x^2 + (-2)x + (-8)
\]
In the format you requested:
- The coefficient of \(x^2\) is **1**.
- The coefficient of \(x\) is **-2**.
- The constant term is **-8**.
Answered by
GPT-4o mini
To simplify the expression \((r - 1)(r^2 - 2r + 3)\), we will use the distributive property, multiplying each term in the first polynomial by each term in the second polynomial.
Here are the steps:
1. Multiply \(r\) by each term in \(r^2 - 2r + 3\):
\[
r \cdot r^2 = r^3
\]
\[
r \cdot (-2r) = -2r^2
\]
\[
r \cdot 3 = 3r
\]
2. Multiply \(-1\) by each term in \(r^2 - 2r + 3\):
\[
-1 \cdot r^2 = -r^2
\]
\[
-1 \cdot (-2r) = 2r
\]
\[
-1 \cdot 3 = -3
\]
Now, combine all these results:
\[
r^3 - 2r^2 + 3r - r^2 + 2r - 3
\]
Next, combine the like terms:
- The \(r^2\) terms: \(-2r^2 - r^2 = -3r^2\)
- The \(r\) terms: \(3r + 2r = 5r\)
Putting it all together, we have:
\[
r^3 - 3r^2 + 5r - 3
\]
Therefore, the final simplified expression can be written as:
\[
r^3 + (-3)r^2 + 5r + (-3)
\]
In the format you requested:
- The coefficient of \(r^3\) is **1**.
- The coefficient of \(r^2\) is **-3**.
- The coefficient of \(r\) is **5**.
- The constant term is **-3**.
Here are the steps:
1. Multiply \(r\) by each term in \(r^2 - 2r + 3\):
\[
r \cdot r^2 = r^3
\]
\[
r \cdot (-2r) = -2r^2
\]
\[
r \cdot 3 = 3r
\]
2. Multiply \(-1\) by each term in \(r^2 - 2r + 3\):
\[
-1 \cdot r^2 = -r^2
\]
\[
-1 \cdot (-2r) = 2r
\]
\[
-1 \cdot 3 = -3
\]
Now, combine all these results:
\[
r^3 - 2r^2 + 3r - r^2 + 2r - 3
\]
Next, combine the like terms:
- The \(r^2\) terms: \(-2r^2 - r^2 = -3r^2\)
- The \(r\) terms: \(3r + 2r = 5r\)
Putting it all together, we have:
\[
r^3 - 3r^2 + 5r - 3
\]
Therefore, the final simplified expression can be written as:
\[
r^3 + (-3)r^2 + 5r + (-3)
\]
In the format you requested:
- The coefficient of \(r^3\) is **1**.
- The coefficient of \(r^2\) is **-3**.
- The coefficient of \(r\) is **5**.
- The constant term is **-3**.
Answered by
GPT-4o mini
To simplify the polynomial expression \((xy - 2)(x^2 + 1)\), we will use the distributive property to multiply each term in the first polynomial by each term in the second polynomial.
Here are the steps:
1. Multiply \(xy\) by each term in \(x^2 + 1\):
\[
xy \cdot x^2 = x^3y
\]
\[
xy \cdot 1 = xy
\]
2. Multiply \(-2\) by each term in \(x^2 + 1\):
\[
-2 \cdot x^2 = -2x^2
\]
\[
-2 \cdot 1 = -2
\]
Now, combine all these results:
\[
x^3y + xy - 2x^2 - 2
\]
This can be arranged in standard form:
\[
x^3y - 2x^2 + xy - 2
\]
Thus, in the format you requested:
- The coefficient of \(x^3y\) is **1**.
- The coefficient of \(x^2\) is **-2**.
- The coefficient of \(xy\) is **1**.
- The constant term is **-2**.
So the simplified expression can be written as:
\[
x^3y + (-2)x^2 + xy + (-2)
\]
Here are the steps:
1. Multiply \(xy\) by each term in \(x^2 + 1\):
\[
xy \cdot x^2 = x^3y
\]
\[
xy \cdot 1 = xy
\]
2. Multiply \(-2\) by each term in \(x^2 + 1\):
\[
-2 \cdot x^2 = -2x^2
\]
\[
-2 \cdot 1 = -2
\]
Now, combine all these results:
\[
x^3y + xy - 2x^2 - 2
\]
This can be arranged in standard form:
\[
x^3y - 2x^2 + xy - 2
\]
Thus, in the format you requested:
- The coefficient of \(x^3y\) is **1**.
- The coefficient of \(x^2\) is **-2**.
- The coefficient of \(xy\) is **1**.
- The constant term is **-2**.
So the simplified expression can be written as:
\[
x^3y + (-2)x^2 + xy + (-2)
\]
Answered by
GPT-4o mini
To determine which option shows that polynomials form a closed system under multiplication, we need to identify which of the given expressions demonstrates that the product of two polynomials is also a polynomial.
1. **Option #1: \(3(2x + 5) = 6x + 15\)**
- This expression represents the multiplication of a polynomial \(2x + 5\) by a constant \(3\). The result is still a polynomial. However, this does not show multiplication between two polynomials.
2. **Option #2: \(8x + 7 = 5x + 3x + 1 + 6\)**
- This expression demonstrates how to combine like terms but does not involve multiplication at all. So, it does not show closure under multiplication.
3. **Option #3: \((2x^2 - 4)(3y + 6) = 6x^2y + 12x^2 - 12y - 24\)**
- This expression involves the multiplication of two polynomials: \(2x^2 - 4\) and \(3y + 6\). The result is also a polynomial, specifically \((2x^2 - 4)(3y + 6)\), confirming that polynomials are closed under multiplication.
4. **Option #4: \(2x^{12} \cdot 3 = 6x\sqrt{}\)**
- This option appears to have formatting issues, but the term \(2x^{12} \cdot 3 = 6x\sqrt{}\) does not show proper polynomial multiplication and likely does not yield a polynomial.
Given this analysis, **Option #3** is the correct option that demonstrates that polynomials form a closed system under multiplication.
1. **Option #1: \(3(2x + 5) = 6x + 15\)**
- This expression represents the multiplication of a polynomial \(2x + 5\) by a constant \(3\). The result is still a polynomial. However, this does not show multiplication between two polynomials.
2. **Option #2: \(8x + 7 = 5x + 3x + 1 + 6\)**
- This expression demonstrates how to combine like terms but does not involve multiplication at all. So, it does not show closure under multiplication.
3. **Option #3: \((2x^2 - 4)(3y + 6) = 6x^2y + 12x^2 - 12y - 24\)**
- This expression involves the multiplication of two polynomials: \(2x^2 - 4\) and \(3y + 6\). The result is also a polynomial, specifically \((2x^2 - 4)(3y + 6)\), confirming that polynomials are closed under multiplication.
4. **Option #4: \(2x^{12} \cdot 3 = 6x\sqrt{}\)**
- This option appears to have formatting issues, but the term \(2x^{12} \cdot 3 = 6x\sqrt{}\) does not show proper polynomial multiplication and likely does not yield a polynomial.
Given this analysis, **Option #3** is the correct option that demonstrates that polynomials form a closed system under multiplication.
Answered by
GPT-4o mini
To simplify the expression \((b + 8)(3b - 6)\), we will use the distributive property (FOIL method):
1. **Multiply the first terms:**
\[
b \cdot 3b = 3b^2
\]
2. **Multiply the outer terms:**
\[
b \cdot (-6) = -6b
\]
3. **Multiply the inner terms:**
\[
8 \cdot 3b = 24b
\]
4. **Multiply the last terms:**
\[
8 \cdot (-6) = -48
\]
Now, combine all these results together:
\[
3b^2 - 6b + 24b - 48
\]
Next, combine the like terms:
\[
-6b + 24b = 18b
\]
Putting it all together, we get:
\[
3b^2 + 18b - 48
\]
Thus, the simplified expression is:
\[
3b^2 + 18b - 48
\]
From the given responses, the correct answer is:
**3b² + 18b - 48**.
1. **Multiply the first terms:**
\[
b \cdot 3b = 3b^2
\]
2. **Multiply the outer terms:**
\[
b \cdot (-6) = -6b
\]
3. **Multiply the inner terms:**
\[
8 \cdot 3b = 24b
\]
4. **Multiply the last terms:**
\[
8 \cdot (-6) = -48
\]
Now, combine all these results together:
\[
3b^2 - 6b + 24b - 48
\]
Next, combine the like terms:
\[
-6b + 24b = 18b
\]
Putting it all together, we get:
\[
3b^2 + 18b - 48
\]
Thus, the simplified expression is:
\[
3b^2 + 18b - 48
\]
From the given responses, the correct answer is:
**3b² + 18b - 48**.
Answered by
GPT-4o mini
To find the product of the polynomials \((x^2y + 2)(x^2 - y)\), we will use the distributive property (FOIL method):
1. **Multiply the first terms:**
\[
x^2y \cdot x^2 = x^4y
\]
2. **Multiply the outer terms:**
\[
x^2y \cdot (-y) = -x^2y^2
\]
3. **Multiply the inner terms:**
\[
2 \cdot x^2 = 2x^2
\]
4. **Multiply the last terms:**
\[
2 \cdot (-y) = -2y
\]
Now, combine all these results:
\[
x^4y - x^2y^2 + 2x^2 - 2y
\]
Thus, the final product of the polynomials \((x^2y + 2)(x^2 - y)\) is:
\[
x^4y - x^2y^2 + 2x^2 - 2y
\]
From the given responses, the correct answer is:
**\(x^4y - x^2y^2 + 2x^2 - 2y\)**.
1. **Multiply the first terms:**
\[
x^2y \cdot x^2 = x^4y
\]
2. **Multiply the outer terms:**
\[
x^2y \cdot (-y) = -x^2y^2
\]
3. **Multiply the inner terms:**
\[
2 \cdot x^2 = 2x^2
\]
4. **Multiply the last terms:**
\[
2 \cdot (-y) = -2y
\]
Now, combine all these results:
\[
x^4y - x^2y^2 + 2x^2 - 2y
\]
Thus, the final product of the polynomials \((x^2y + 2)(x^2 - y)\) is:
\[
x^4y - x^2y^2 + 2x^2 - 2y
\]
From the given responses, the correct answer is:
**\(x^4y - x^2y^2 + 2x^2 - 2y\)**.