First slide

Theory of equations

Question

If $a, b, c, d \in R$ then the equation $(x^{2} + a x - 3 b) (x^{2} - c x + b) (x^{2} - d x + 2 b) = 0$

Moderate

A

6 real roots
B

at least 2 real roots
C

4 real roots
D

3 real roots

Solution

The discriminants of the given equations are
$\begin{array}{l} D_{1} = a^{2} + 12 b \\ D_{2} = c^{2} - 4 b and D_{3} = d^{2} - 8 b \\ ∴ D_{1} + D_{2} + D_{3} = a^{2} + c^{2} + d^{2} \geq 0 \end{array}$
Hence, at least one of $D_{1}, D_{2}, D_{3}$ is non-negative. Therefore,
the equation has at least two real roots.

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