18. \( x^{6}-7 x^{3}-8 \)

The polynomial \( x^{6}-7x^{3}-8 \) can be transformed by substituting \( y = x^3 \), simplifying it to \( y^2 - 7y - 8 \). This quadratic expression can be factored or solved using the quadratic formula, revealing two potential real solutions for \( y \). Subsequently, you can find the values of \( x \) by taking the cube root of those solutions. One common mistake when working with higher-degree polynomials is neglecting to check for rational roots. The Rational Root Theorem can guide you in identifying possible rational solutions by considering the factors of the constant term (in this case, -8) divided by the factors of the leading coefficient (which is 1). Always test these potential roots first to save time and avoid unnecessary calculations.