Let \(P(x)=x^{3}-3b^{2}x+16\). If \(P(|x|)\) has exactly two real roots, then the area bounded by \(y=P(x)\) and \(x\)-axis is?

The answer is infinite. Because if \(ω\) is a root of \(P(|x|)\) then \(-\omega\) is also a root of \(P(|x|)\). This means \(P(x)\) can have only one real root either \(\omega\) or \(-\omega\).This means that \(P(x)\) intersects the \(x\)-axis once. So area bounded by it and \(x-axis\) must be infinite.

This question was posted by me . And Pi Han goh and Calvin lin reported it and deleted it saying answer is not always infinity. But answer is always infinite .

Please tell me if I am right .

## Comments

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TopNewestI disagree with " \( P(|x| \) has exactly two real roots means that \( P(x) \) can have only one real root". What is true is

As an explicit example, we could have \( P( -10 ) = P(-4) = P(6) = 0 \). Then if we solve for \( P(|x|) = 0 \), we only get the roots \( x = 6, -6 \).

Do you see the error in your argument? When you said "because if \( \omega \) is a root", what should that actually be?

Please be mindful of your title. I generally have a very good reason if I believe that something is wrong. It would be better for you to ask "Why was this problem deleted" instead of claiming that "Calvin Lin solved this wrongly". Of course, I do admit that there are times that I solve the problem / resolve the report incorrectly, because I'm speed reading through it / missed a condition. – Calvin Lin Staff · 1 year, 2 months ago

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– Shivam Jadhav · 1 year, 2 months ago

Sir but here product of roots is \(-16\) . So the case stated by you doesn't satisfy.Log in to reply

Note that there is no explicit counter example to the entire problem, because there are no polynomials which satisfy the conditions that you stated, namely

1. \( P(x) = x^3 - 3b^2 x + 16 \)

2. \( P(|x|) \) has 2 real roots counted with multiplicity.

Reasoning: From condition 1, since \( P(0) > 0 \), and we have a leading coefficient of 1, the polynomial either has 2 positive real roots (with multiplicity) or 0 positive real roots. As explained above, condition 2 implies that there is exactly 1 positive real root.

Thus, that's another reason for deletion, since your problem is about the empty set. – Calvin Lin Staff · 1 year, 2 months ago

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– Shivam Jadhav · 1 year, 2 months ago

That's right but considering the question as logical not the calculations one the question can be valid .Log in to reply

– Calvin Lin Staff · 1 year, 2 months ago

How is the question valid? For what value of \(b\) are your conditions satisfied?Log in to reply

Substitute \(b = \pm 2\), the answer is 24. – Pi Han Goh · 1 year, 2 months ago

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– Shivam Jadhav · 1 year, 2 months ago

When \(b=2,-2\) \(P(|x|)\) has three real roots \(2,2,-2\) and as \(2\) is a repeated root we consider that \(P(|x|)\) has three real roots with \(2\) as the repeated root. And in question it is strictly mentioned that \(P(|x|)\) has exactly two real roots. So please notice this and tell that whether I am right or wrong .Log in to reply

And are you counting complex roots as well? If so, you need to make it explicit. – Pi Han Goh · 1 year, 2 months ago

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– Shivam Jadhav · 1 year, 2 months ago

We are taking only real roots . It's mentioned in the questionLog in to reply

– Pi Han Goh · 1 year, 2 months ago

Then why is \(\omega \), a third root of unity in your solution?Log in to reply

– Shivam Jadhav · 1 year, 2 months ago

\(ω\) is a variableLog in to reply

– Pi Han Goh · 1 year, 2 months ago

So what is the value of \(b\) that gives the answer of infinity?Log in to reply

@Pi Han Goh @Calvin Lin – Shivam Jadhav · 1 year, 2 months ago

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