Show that p³q + q³r + r³p is a constant

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In summary, the conversation discusses the statement "p³q + q³r + r³p is a constant" and how to prove it. It is asking to show that the expression always evaluates to the same value, regardless of the values of p, q, and r. This can be proven by using the properties of exponents and algebraic manipulation to simplify the expression to a single, constant value. An example is provided to further understand the statement. There is no specific method or formula to prove this, but the key is to manipulate the expression to a constant value. Proving this statement is significant as it demonstrates a fundamental relationship between the variables p, q, and r.
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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.
 
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  • #2
Re: Show that p³q+q³r+r³p is a constant

Here is my solution:

Given $p+q+r = 0$ and $pq+qr+rp = -3$ Now Let $pqr = k$

Now form an cubic equation whose roots are $x = p\;,q\;,r$

$x^3-(p+q+r)x^2+(pq+qr+rp)x-pqr = 0$

$x^3-3x-k=0\Rightarrow x^3 = 3x+k$

Now If $x = p$ is a root of given equation, Then $p^3 = 3p+k$

Similarly If $x = q$ is a root of given equation, Then $q^3 = 3q+k$

Similarly If $x = r$ is a root of given equation, Then $r^3 = 3r+k$

So $p^3q = (3p+k)q = 3pq+kq$

Similarly $q^3r = (3q+k)r = 3pr+kr$

Similarly $r^3p = (3r+k)p = 3rp+kp$

Now Add all These, we get $p^3q+q^3r+r^3p = 3(pq+qr+rp)+(p+q+r)k = -9+0 = -9$(Constant)
 
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  • #3
Re: Show that p³q+q³r+r³p is a constant

Not elegant but different approach

P = - (q+r)

So p^3 q = - p^2 q^2 – p^2 q r
Similarly q^3 r = q^2 r^2 – q^2 pr
r^3 p = r^2 p^2 – r^2 pr
add all 3 to get
p^3 q + q^3 r + r^3 p = - (p^2q^2 + q^2 r^2 + r^2 p^2) – pqr(p+q + r)
= - (p^2q^2 + q^2 r^2 + r^2 p^2) .. (1)
Now as we have p^2 q^2 + q^2 r^2 + r^2 p^2 above we square
pq+pr+qr=−3 to get
p^2q^2 + p^2 r^2 + q^2 r ^2 + 2p^2qr + 2r^2qp + 2 q^pr = 9
or p^2q^2 + p^2 r^2 + q^2 r ^2 + 2pqr(p + r + q) = 9
or p^2q^2 + p^2 r^2 + q^2 r ^2 = 9 ... (2)
from (1) and (2) p^3 q + q^3 r + r^3 p = - 9
 
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FAQ: Show that p³q + q³r + r³p is a constant

What is the statement "p³q + q³r + r³p is a constant" asking to be proved?

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.

How can I prove that p³q + q³r + r³p is a constant?

You can prove this by using the properties of exponents and algebraic manipulation to show that the expression simplifies to a single, constant value.

Can you provide an example to help understand this statement?

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.

Is there a specific method or formula to prove that this expression is a constant?

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.

What is the significance of proving that p³q + q³r + r³p is a constant?

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.

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