Can All Roots of a Quartic Polynomial Be Real?

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SUMMARY

The discussion centers on the quartic polynomial equation $ax^4 + bx^3 + x^2 + x + 1 = 0$, where $a$ and $b$ are real numbers with $a \neq 0$. It is established that not all roots of this polynomial can be real. The proof involves analyzing the behavior of the polynomial and applying the Fundamental Theorem of Algebra, demonstrating that at least one root must be complex when $a$ is non-zero.

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  • Understanding of polynomial equations and their roots
  • Familiarity with the Fundamental Theorem of Algebra
  • Knowledge of quartic functions and their characteristics
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Let $a$ and $b$ be real numbers such that $a\ne 0$. Prove that not all the roots of $ax^4+bx^3+x^2+x+1=0$ can be real.
 
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Between each pair of real roots of a polynomial there must be a root of the derivative.

Let $x_1,x_2,x_3,x_4$ be the roots of $ax^4+bx^3+x^2+x+1$. Replacing $x$ by $\frac1x$, it follows that $\frac1{x_1},\frac1{x_2},\frac1{x_3},\frac1{x_4}$ are the roots of $p(x) = x^4 + x^3 + x^2 + bx + a$. The second derivative of $p(x)$ is $p''(x) = 12x^2 + 6x + 2$, which has no real roots. So $p'(x)$ can have only one real root, and $p(x)$ has at most two real roots. Therefore at most two of $x_1,x_2,x_3,x_4$ are real.
 

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