3B. Find all values of the real number $k$ for which the expression

$$
\sin ^{6} x+\cos ^{6} x+k\left(\sin ^{4} x+\cos ^{4} x\right)
$$

does not depend on $x$.
Solution. We have

$$
\begin{aligned}
\sin ^{6} x+\cos ^{6} x+ & k\left(\sin ^{4} x+\cos ^{4} x\right)= \\
= & \left(\sin ^{2} x+\cos ^{2} x\right)\left(\sin ^{4} x-\sin ^{2} x \cos ^{2} x+\cos ^{4} x\right) \\
& \quad+k\left(\left(\sin ^{2} x+\cos ^{2} x\right)^{2}-2 \sin ^{2} x \cos ^{2} x\right) \\
= & \left(\sin ^{2} x+\cos ^{2} x\right)^{2}-3 \sin ^{2} x \cos ^{2} x+k\left(1-2 \sin ^{2} x \cos ^{2} x\right) \\
= & 1-\frac{3}{4} \sin ^{2} 2 x+k\left(1-\frac{1}{2} \sin ^{2} 2 x\right)=k+1-\frac{3+2 k}{4} \sin ^{2} 2 x
\end{aligned}
$$

The expression does not depend on $x$ if $3+2 k=0$, i.e., for $k=-\frac{3}{2}$.