Algebra — FAQ

Graphically, the V-shape of $|f(x)|$ intersects the horizontal line $y=k$ at two points. The inequality holds for the parts of the 'V' that are above the line, which are the two 'arms' extending outwards, leading to two separate regions.

It means a quadratic expression that cannot be factorised using real numbers. This occurs when its discriminant, $b^2-4ac$, is negative. For example, in $x^2+4$, the discriminant is $0^2 - 4(1)(4) = -16 < 0$. Its roots are complex.

The expansion is an infinite geometric series in disguise. For an infinite series to converge to a finite sum, the common ratio must have a magnitude less than 1. In the binomial expansion, the terms behave like a geometric progression with ratio related to $x$. If $|x| \ge 1$, the terms get larger and larger, so the sum diverges to infinity and is not a useful approximation.

If $x=2$ is a root, then $(x-2)$ is a factor. You can use polynomial long division to divide the original cubic by $(x-2)$. The result will be a quadratic expression, which you can then solve using the quadratic formula or factorisation to find the remaining two roots.

No. The Complex Conjugate Root Theorem guarantees that complex roots always come in pairs (a root and its conjugate). Therefore, a polynomial with real coefficients must have an even number of complex roots (which could be zero).