Show that p² + p - 6 is equal to q or r.

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In summary, we have shown that $p^2+p-6$ is a root of the polynomial x^3-9x+9 and must be either equal to $q$ or $r$. This is due to the fact that $p$ is a root of x^3-9x+9, making $p^2+p-6$ a root as well since $p^2+p-6 \ne p$. Thus, $p^2+p-6$ must be equal to either $q$ or $r$.
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Let $p$, $q$ and $r$ be roots of polynomial \(\displaystyle x^3-9x+9=0\). Show that \(\displaystyle p^2+p-6\) is equal to $q$ or $r$.
 
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anemone said:
Let $p$, $q$ and $r$ be roots of polynomial \(\displaystyle x^3-9x+9=0\). Show that \(\displaystyle p^2+p-6\) is equal to $q$ or $r$.

It is sufficient to show that \(\displaystyle p^2+p-6\) is a root of \(\displaystyle x^3-9x+9\) different from $p$. We may as well also note that \(\displaystyle x^3-9x+9\) has three distinct real roots.

Well substitute the first into the second and we get:

$$(p^2+p-6)^3-9(p^2+p-6)+9=(p^3-9p+9)(p^3+3p^2-6p-17)$$

but as $p$ is a root of \(\displaystyle x^3-9x+9\) we have:

$$(p^2+p-6)^3-9(p^2+p-6)+9=(p^3-9p+9)(p^3+3p^2-6p-17)=0$$

That is $p^2+p-6$ is a root of the cubic, and as neither $\pm \sqrt{6}$ is a root of the cubic $p^2+p-6 \ne p$ hence $p^2+p-6$ must be equal to $q$ or $r$.

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FAQ: Show that p² + p - 6 is equal to q or r.

What does "p² + p - 6 is equal to q or r" mean?

This statement refers to a mathematical expression where p is a variable and q and r are either constants or variables. It is asking to show that the expression is equal to either q or r.

How do you solve "p² + p - 6 = q or r"?

To solve this expression, you can use the quadratic formula or factor the expression to find the values of p that make the equation true for both q and r.

Can you provide an example of solving "p² + p - 6 = q or r"?

Sure, let's say q = 5 and r = -3. By factoring the expression, we get (p+3)(p-2) = 0, which means that p = -3 or p = 2. Substituting these values into the original expression gives us -3² + (-3) - 6 = 5, and 2² + 2 - 6 = -3, satisfying the condition that the expression is equal to either q or r.

What is the significance of showing "p² + p - 6 is equal to q or r"?

By showing that this expression is equal to either q or r, we are proving that there are multiple values of p that can satisfy the equation. This can have implications in various fields of science and mathematics, such as finding solutions to equations or analyzing patterns in data.

Are there any real-world applications of "p² + p - 6 = q or r"?

Yes, this type of expression can be used in various fields such as physics, engineering, and economics. For example, in physics, it can be used to solve equations describing motion, and in economics, it can be used to analyze supply and demand models.

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