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anemone
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Show that for all real numbers $p, q, r$ such that $p+q+r=0$ and $pq+pr+qr=-3$, the expression $p^3q+q^3r+r^3p$ is a constant.
The statement is asking to prove that the expression p³q + q³r + r³p always evaluates to the same value, regardless of the values of p, q, and r.
You can prove this by using the properties of exponents and algebraic manipulation to show that the expression simplifies to a single, constant value.
Sure, for example, if p = 2, q = 3, and r = 4, then the expression p³q + q³r + r³p evaluates to (2³)(3) + (3³)(4) + (4³)(2) = 8(3) + 27(4) + 64(2) = 24 + 108 + 128 = 260. This means that regardless of the specific values chosen for p, q, and r, the expression will always evaluate to 260.
No, there is no specific method or formula. The key is to manipulate the expression using the properties of exponents and algebra to simplify it to a single, constant value.
This statement is significant because it shows that no matter what values are chosen for p, q, and r, the expression will always evaluate to the same value. This can be useful in various mathematical and scientific applications, as it demonstrates a fundamental relationship between these variables.