Show x³y+y³z+z³x is a constant

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In summary, the formula for "Show x³y+y³z+z³x is a constant" is (x+y+z)(x²+y²+z²-xy-yz-zx). To prove that x³y+y³z+z³x is a constant, you can use algebraic manipulation or the properties of symmetric polynomials. Its significance lies in its independence from the values of x, y, and z, making it useful in solving equations and understanding relationships between variables. An example of x³y+y³z+z³x being a constant is in the expansion of (x+y+z)⁴. It is related to symmetric polynomials and invariants.
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Show that for all real numbers $x,\,y,\,z$ such that $x+y+z=0$ and $xy+yz+zx=-3$, the expression $x^3y+y^3z+z^3x$ is a constant.
 
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anemone said:
Show that for all real numbers $x,\,y,\,z$ such that $x+y+z=0$ and $xy+yz+zx=-3$, the expression $x^3y+y^3z+z^3x$ is a constant.
We have (given)
$x+y+z=0 \cdots(1)$
$xy+yz+zx=-3\cdots(2)$
From (1)
$(x+y+z)^2 = x^2 + y^2 + z^2 + 2(xy +yz + zx) = 0$
or $x^2 + y^2 + z^2 + 2(-3) = 0$
or $x^2 + y^2 + z^2 = 6\cdots(3)$
Further from (1)
$x+y = -z\cdots(4)$
$y+z = -x\cdots(5)$
$z+x = -y\cdots(6)$

Now let us prove that
$x^3y+ y^3 z + z^3 x = xy^3 + yz^3 + zx^3\cdots(7)$

to prove the same
$x^3y+ y^3 z + z^3 x - (xy^3 + yz^3 + zx^3)$
$= (x^3y - xy^3) + (y^3z - yz^3) + (z^3 x - zx^3)$
$= xy(x^2 - y^2) + yz(y^2 - z^2) + zx(z^2 - x^2)$
$=xy(x+y)(x-y) + yz(y+z)( y-z) + zx(z+x)(z-x)$
$=xy(-z)(x-y) + yz(-x) (y-z) + xz(-y) (z-x)$ using (4), (5), (6)
$= - xyz(x-y) - xyz(y-z) - xyz(z-x)$
= 0so (7) is true

Now $(x^2 + y^2 + z^2)(xy + yz + zx) = 6 * (-3) $ putting values from above
Or $x^3y + x^2yz + zx^3 + xy^3 + y^3 z + y^2zx + z^2yx + yz^3 + z^3 x = 18$
or $(x^3y + y^3 z + z^3x) + (xy^3 + yz^3 + zx^3 ) + (x^2yz + xy^2z + xyz^2) = - 18$
or $(x^3y + y^3 z + z^3x) + (x^3y + y^3z + z^3x ) + (x^2yz + xy^2z + xyz^2) = - 18$ (from (7)
or $2(x^3y + y^3 z + z^3x) + xyz(x+y+z) = - 18$
or $2(x^3y + y^3 z + z^3x) + xyz. 0 = - 18$ from (1)
or $2(x^3y + y^3 z + z^3x)= - 18$ from (1)
or $(x^3y + y^3 z + z^3x) = - 9$

Which is a constant

Hence proved
 

FAQ: Show x³y+y³z+z³x is a constant

1. What is the significance of "Show x³y+y³z+z³x is a constant" in scientific research?

The expression x³y+y³z+z³x is a constant is a fundamental concept in mathematics and physics. It is often used to represent conservation laws, where the quantity represented by the expression remains constant regardless of changes in other variables. In scientific research, this concept is crucial in understanding the behavior of physical systems and making predictions.

2. How can one prove that x³y+y³z+z³x is a constant?

To prove that x³y+y³z+z³x is a constant, one can use mathematical techniques such as differentiation and integration. By taking the derivative of the expression with respect to any of the variables, we can show that it is equal to zero, indicating that the expression is constant. Alternatively, we can also use algebraic manipulations to show that the expression remains unchanged, regardless of the values of x, y, and z.

3. What is the geometric interpretation of x³y+y³z+z³x is a constant?

The expression x³y+y³z+z³x can be interpreted geometrically as a surface in three-dimensional space. The constant value of the expression represents a level surface, where all points on the surface have the same value. This surface can take on different shapes depending on the specific values of x, y, and z, but the constant value remains the same.

4. In what real-world applications is x³y+y³z+z³x is a constant used?

The concept of x³y+y³z+z³x is a constant is widely used in various fields of science and engineering. It is commonly used in physics to represent conservation laws, such as the conservation of energy or momentum. In chemistry, it is used to describe the behavior of chemical reactions. It is also used in economics and finance to model the relationship between different variables.

5. Can x³y+y³z+z³x be a constant if x, y, and z are not constant?

Yes, x³y+y³z+z³x can be a constant even if x, y, and z are not constant. This is because the expression represents a relationship between the variables, and the constant value indicates that this relationship remains the same regardless of changes in the individual variables. In other words, the values of x, y, and z can change, but their combined effect on the expression remains constant.

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