At Westonci.ca, we provide clear, reliable answers to all your questions. Join our vibrant community and get the solutions you need. Get expert answers to your questions quickly and accurately from our dedicated community of professionals. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
To show that if [tex]\( x + y + z = 0 \)[/tex], then [tex]\( x^3 + y^3 + z^3 = 3xyz \)[/tex], let's proceed with the following steps:
### Step 1: Use the Identity for the Sum of Cubes
We start with a well-known algebraic identity for the sum of cubes:
[tex]\[ x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx) \][/tex]
### Step 2: Substitute the Given Condition
Given that [tex]\( x + y + z = 0 \)[/tex], we substitute this into the identity:
[tex]\[ x^3 + y^3 + z^3 - 3xyz = (0)(x^2 + y^2 + z^2 - xy - yz - zx) \][/tex]
### Step 3: Simplify the Equation
Since any number multiplied by 0 is 0, the right-hand side of the equation becomes 0:
[tex]\[ x^3 + y^3 + z^3 - 3xyz = 0 \][/tex]
### Step 4: Rearrange to Show the Desired Result
Rearranging the equation, we get:
[tex]\[ x^3 + y^3 + z^3 = 3xyz \][/tex]
### Verification with Example Values
To confirm this identity, let's take specific values of [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex] that satisfy [tex]\( x + y + z = 0 \)[/tex]. For instance, let [tex]\( x = 1 \)[/tex], [tex]\( y = -2 \)[/tex], and [tex]\( z = 1 \)[/tex]:
[tex]\[ x + y + z = 1 - 2 + 1 = 0 \][/tex]
This satisfies the given condition. Now, we calculate both sides of the equation:
1. Calculate the left side:
[tex]\[ x^3 + y^3 + z^3 = 1^3 + (-2)^3 + 1^3 = 1 - 8 + 1 = -6 \][/tex]
2. Calculate the right side:
[tex]\[ 3xyz = 3 \cdot 1 \cdot (-2) \cdot 1 = 3 \cdot (-2) = -6 \][/tex]
Both sides of the equation equal [tex]\(-6\)[/tex], confirming that the relation [tex]\( x^3 + y^3 + z^3 = 3xyz \)[/tex] holds true for these values.
Thus, we have shown that if [tex]\( x + y + z = 0 \)[/tex], then [tex]\( x^3 + y^3 + z^3 = 3xyz \)[/tex].
### Step 1: Use the Identity for the Sum of Cubes
We start with a well-known algebraic identity for the sum of cubes:
[tex]\[ x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx) \][/tex]
### Step 2: Substitute the Given Condition
Given that [tex]\( x + y + z = 0 \)[/tex], we substitute this into the identity:
[tex]\[ x^3 + y^3 + z^3 - 3xyz = (0)(x^2 + y^2 + z^2 - xy - yz - zx) \][/tex]
### Step 3: Simplify the Equation
Since any number multiplied by 0 is 0, the right-hand side of the equation becomes 0:
[tex]\[ x^3 + y^3 + z^3 - 3xyz = 0 \][/tex]
### Step 4: Rearrange to Show the Desired Result
Rearranging the equation, we get:
[tex]\[ x^3 + y^3 + z^3 = 3xyz \][/tex]
### Verification with Example Values
To confirm this identity, let's take specific values of [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex] that satisfy [tex]\( x + y + z = 0 \)[/tex]. For instance, let [tex]\( x = 1 \)[/tex], [tex]\( y = -2 \)[/tex], and [tex]\( z = 1 \)[/tex]:
[tex]\[ x + y + z = 1 - 2 + 1 = 0 \][/tex]
This satisfies the given condition. Now, we calculate both sides of the equation:
1. Calculate the left side:
[tex]\[ x^3 + y^3 + z^3 = 1^3 + (-2)^3 + 1^3 = 1 - 8 + 1 = -6 \][/tex]
2. Calculate the right side:
[tex]\[ 3xyz = 3 \cdot 1 \cdot (-2) \cdot 1 = 3 \cdot (-2) = -6 \][/tex]
Both sides of the equation equal [tex]\(-6\)[/tex], confirming that the relation [tex]\( x^3 + y^3 + z^3 = 3xyz \)[/tex] holds true for these values.
Thus, we have shown that if [tex]\( x + y + z = 0 \)[/tex], then [tex]\( x^3 + y^3 + z^3 = 3xyz \)[/tex].
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.