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Sagot :
Certainly! Let's expand each of the given expressions step-by-step.
### 1. Expanding [tex]\((a + 2b + c)^2\)[/tex]
To expand [tex]\((a + 2b + c)^2\)[/tex], we can use the formula [tex]\((x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2xz + 2yz\)[/tex]:
[tex]\[ (a + 2b + c)^2 = a^2 + (2b)^2 + c^2 + 2(a \cdot 2b) + 2(a \cdot c) + 2(2b \cdot c) \][/tex]
[tex]\[ = a^2 + 4b^2 + c^2 + 4ab + 2ac + 4bc \][/tex]
So, the expanded form is:
[tex]\[ a^2 + 4ab + 2ac + 4b^2 + 4bc + c^2 \][/tex]
### 2. Expanding [tex]\((2a - 3b - c)^2\)[/tex]
For [tex]\((2a - 3b - c)^2\)[/tex], we use the same formula:
[tex]\[ (2a - 3b - c)^2 = (2a)^2 + (-3b)^2 + (-c)^2 + 2(2a \cdot -3b) + 2(2a \cdot -c) + 2(-3b \cdot -c) \][/tex]
[tex]\[ = 4a^2 + 9b^2 + c^2 - 12ab - 4ac + 6bc \][/tex]
So, the expanded form is:
[tex]\[ 4a^2 - 12ab - 4ac + 9b^2 + 6bc + c^2 \][/tex]
### 3. Expanding [tex]\((-3x + y + z)^2\)[/tex]
For [tex]\((-3x + y + z)^2\)[/tex]:
[tex]\[ (-3x + y + z)^2 = (-3x)^2 + y^2 + z^2 + 2(-3x \cdot y) + 2(-3x \cdot z) + 2(y \cdot z) \][/tex]
[tex]\[ = 9x^2 + y^2 + z^2 - 6xy - 6xz + 2yz \][/tex]
So, the expanded form is:
[tex]\[ 9x^2 - 6xy - 6xz + y^2 + 2yz + z^2 \][/tex]
### 4. Expanding [tex]\((m + 2n - 5p)^2\)[/tex]
For [tex]\((m + 2n - 5p)^2\)[/tex]:
[tex]\[ (m + 2n - 5p)^2 = m^2 + (2n)^2 + (-5p)^2 + 2(m \cdot 2n) + 2(m \cdot -5p) + 2(2n \cdot -5p) \][/tex]
[tex]\[ = m^2 + 4n^2 + 25p^2 + 4mn - 10mp - 20np \][/tex]
So, the expanded form is:
[tex]\[ m^2 + 4mn - 10mp + 4n^2 - 20np + 25p^2 \][/tex]
### 5. Expanding [tex]\((2 + x - 2y)^2\)[/tex]
For [tex]\((2 + x - 2y)^2\)[/tex]:
[tex]\[ (2 + x - 2y)^2 = 2^2 + x^2 + (-2y)^2 + 2(2 \cdot x) + 2(2 \cdot -2y) + 2(x \cdot -2y) \][/tex]
[tex]\[ = 4 + x^2 + 4y^2 + 4x - 8y - 4xy \][/tex]
So, the expanded form is:
[tex]\[ x^2 - 4xy + 4x + 4y^2 - 8y + 4 \][/tex]
### 6. Expanding [tex]\((a^2 + b^2 + c^2)^2\)[/tex]
For [tex]\((a^2 + b^2 + c^2)^2\)[/tex]:
[tex]\[ (a^2 + b^2 + c^2)^2 = (a^2)^2 + (b^2)^2 + (c^2)^2 + 2(a^2 \cdot b^2) + 2(a^2 \cdot c^2) + 2(b^2 \cdot c^2) \][/tex]
[tex]\[ = a^4 + b^4 + c^4 + 2a^2b^2 + 2a^2c^2 + 2b^2c^2 \][/tex]
So, the expanded form is:
[tex]\[ a^4 + 2a^2b^2 + 2a^2c^2 + b^4 + 2b^2c^2 + c^4 \][/tex]
### 7. Expanding [tex]\(\left(\frac{x}{y} + \frac{y}{z} + \frac{z}{x}\right)^2\)[/tex]
For [tex]\(\left(\frac{x}{y} + \frac{y}{z} + \frac{z}{x}\right)^2\)[/tex]:
[tex]\[ \left(\frac{x}{y} + \frac{y}{z} + \frac{z}{x}\right)^2 = \left(\frac{x}{y}\right)^2 + \left(\frac{y}{z}\right)^2 + \left(\frac{z}{x}\right)^2 + 2\left(\frac{x}{y} \cdot \frac{y}{z}\right) + 2\left(\frac{x}{y} \cdot \frac{z}{x}\right) + 2\left(\frac{y}{z} \cdot \frac{z}{x}\right) \][/tex]
[tex]\[ = \frac{x^2}{y^2} + \frac{y^2}{z^2} + \frac{z^2}{x^2} + 2 \left(\frac{x}{z} + \frac{y}{x} + \frac{z}{y} \right) \][/tex]
So, the expanded form is:
[tex]\[ \frac{x^2}{y^2} + \frac{y^2}{z^2} + \frac{z^2}{x^2} + 2 \frac{x}{z} + 2 \frac{y}{x} + 2 \frac{z}{y} \][/tex]
Thus, the expanded forms of the given expressions are:
1. [tex]\(a^2 + 4ab + 2ac + 4b^2 + 4bc + c^2\)[/tex]
2. [tex]\(4a^2 - 12ab - 4ac + 9b^2 + 6bc + c^2\)[/tex]
3. [tex]\(9x^2 - 6xy - 6xz + y^2 + 2yz + z^2\)[/tex]
4. [tex]\(m^2 + 4mn - 10mp + 4n^2 - 20np + 25p^2\)[/tex]
5. [tex]\(x^2 - 4xy + 4x + 4y^2 - 8y + 4\)[/tex]
6. [tex]\(a^4 + 2a^2b^2 + 2a^2c^2 + b^4 + 2b^2c^2 + c^4\)[/tex]
7. [tex]\(\frac{x^2}{y^2} + \frac{y^2}{z^2} + \frac{z^2}{x^2} + 2 \frac{x}{z} + 2 \frac{y}{x} + 2 \frac{z}{y}\)[/tex]
### 1. Expanding [tex]\((a + 2b + c)^2\)[/tex]
To expand [tex]\((a + 2b + c)^2\)[/tex], we can use the formula [tex]\((x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2xz + 2yz\)[/tex]:
[tex]\[ (a + 2b + c)^2 = a^2 + (2b)^2 + c^2 + 2(a \cdot 2b) + 2(a \cdot c) + 2(2b \cdot c) \][/tex]
[tex]\[ = a^2 + 4b^2 + c^2 + 4ab + 2ac + 4bc \][/tex]
So, the expanded form is:
[tex]\[ a^2 + 4ab + 2ac + 4b^2 + 4bc + c^2 \][/tex]
### 2. Expanding [tex]\((2a - 3b - c)^2\)[/tex]
For [tex]\((2a - 3b - c)^2\)[/tex], we use the same formula:
[tex]\[ (2a - 3b - c)^2 = (2a)^2 + (-3b)^2 + (-c)^2 + 2(2a \cdot -3b) + 2(2a \cdot -c) + 2(-3b \cdot -c) \][/tex]
[tex]\[ = 4a^2 + 9b^2 + c^2 - 12ab - 4ac + 6bc \][/tex]
So, the expanded form is:
[tex]\[ 4a^2 - 12ab - 4ac + 9b^2 + 6bc + c^2 \][/tex]
### 3. Expanding [tex]\((-3x + y + z)^2\)[/tex]
For [tex]\((-3x + y + z)^2\)[/tex]:
[tex]\[ (-3x + y + z)^2 = (-3x)^2 + y^2 + z^2 + 2(-3x \cdot y) + 2(-3x \cdot z) + 2(y \cdot z) \][/tex]
[tex]\[ = 9x^2 + y^2 + z^2 - 6xy - 6xz + 2yz \][/tex]
So, the expanded form is:
[tex]\[ 9x^2 - 6xy - 6xz + y^2 + 2yz + z^2 \][/tex]
### 4. Expanding [tex]\((m + 2n - 5p)^2\)[/tex]
For [tex]\((m + 2n - 5p)^2\)[/tex]:
[tex]\[ (m + 2n - 5p)^2 = m^2 + (2n)^2 + (-5p)^2 + 2(m \cdot 2n) + 2(m \cdot -5p) + 2(2n \cdot -5p) \][/tex]
[tex]\[ = m^2 + 4n^2 + 25p^2 + 4mn - 10mp - 20np \][/tex]
So, the expanded form is:
[tex]\[ m^2 + 4mn - 10mp + 4n^2 - 20np + 25p^2 \][/tex]
### 5. Expanding [tex]\((2 + x - 2y)^2\)[/tex]
For [tex]\((2 + x - 2y)^2\)[/tex]:
[tex]\[ (2 + x - 2y)^2 = 2^2 + x^2 + (-2y)^2 + 2(2 \cdot x) + 2(2 \cdot -2y) + 2(x \cdot -2y) \][/tex]
[tex]\[ = 4 + x^2 + 4y^2 + 4x - 8y - 4xy \][/tex]
So, the expanded form is:
[tex]\[ x^2 - 4xy + 4x + 4y^2 - 8y + 4 \][/tex]
### 6. Expanding [tex]\((a^2 + b^2 + c^2)^2\)[/tex]
For [tex]\((a^2 + b^2 + c^2)^2\)[/tex]:
[tex]\[ (a^2 + b^2 + c^2)^2 = (a^2)^2 + (b^2)^2 + (c^2)^2 + 2(a^2 \cdot b^2) + 2(a^2 \cdot c^2) + 2(b^2 \cdot c^2) \][/tex]
[tex]\[ = a^4 + b^4 + c^4 + 2a^2b^2 + 2a^2c^2 + 2b^2c^2 \][/tex]
So, the expanded form is:
[tex]\[ a^4 + 2a^2b^2 + 2a^2c^2 + b^4 + 2b^2c^2 + c^4 \][/tex]
### 7. Expanding [tex]\(\left(\frac{x}{y} + \frac{y}{z} + \frac{z}{x}\right)^2\)[/tex]
For [tex]\(\left(\frac{x}{y} + \frac{y}{z} + \frac{z}{x}\right)^2\)[/tex]:
[tex]\[ \left(\frac{x}{y} + \frac{y}{z} + \frac{z}{x}\right)^2 = \left(\frac{x}{y}\right)^2 + \left(\frac{y}{z}\right)^2 + \left(\frac{z}{x}\right)^2 + 2\left(\frac{x}{y} \cdot \frac{y}{z}\right) + 2\left(\frac{x}{y} \cdot \frac{z}{x}\right) + 2\left(\frac{y}{z} \cdot \frac{z}{x}\right) \][/tex]
[tex]\[ = \frac{x^2}{y^2} + \frac{y^2}{z^2} + \frac{z^2}{x^2} + 2 \left(\frac{x}{z} + \frac{y}{x} + \frac{z}{y} \right) \][/tex]
So, the expanded form is:
[tex]\[ \frac{x^2}{y^2} + \frac{y^2}{z^2} + \frac{z^2}{x^2} + 2 \frac{x}{z} + 2 \frac{y}{x} + 2 \frac{z}{y} \][/tex]
Thus, the expanded forms of the given expressions are:
1. [tex]\(a^2 + 4ab + 2ac + 4b^2 + 4bc + c^2\)[/tex]
2. [tex]\(4a^2 - 12ab - 4ac + 9b^2 + 6bc + c^2\)[/tex]
3. [tex]\(9x^2 - 6xy - 6xz + y^2 + 2yz + z^2\)[/tex]
4. [tex]\(m^2 + 4mn - 10mp + 4n^2 - 20np + 25p^2\)[/tex]
5. [tex]\(x^2 - 4xy + 4x + 4y^2 - 8y + 4\)[/tex]
6. [tex]\(a^4 + 2a^2b^2 + 2a^2c^2 + b^4 + 2b^2c^2 + c^4\)[/tex]
7. [tex]\(\frac{x^2}{y^2} + \frac{y^2}{z^2} + \frac{z^2}{x^2} + 2 \frac{x}{z} + 2 \frac{y}{x} + 2 \frac{z}{y}\)[/tex]
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