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Polynomials & Complex Numbers

Chapter 1 · Quadratics & Polynomials

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Standard form
y=ax2+bx+c(a≠0)y = ax^2 + bx + c \quad (a \neq 0)
Vertex form
y=a(x−h)2+ky = a(x-h)^2 + k
Vertex at (h,k)(h, k).
Factored form
y=a(x−p)(x−q)y = a(x-p)(x-q)
Roots at x=p,qx = p, q.
Axis of symmetry / vertex x
x=−b2ax = -\frac{b}{2a}
Quadratic formula
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
Discriminant
Δ=b2−4ac\Delta = b^2 - 4ac
>0>0: 2 roots · =0=0: 1 repeated · <0<0: none real.
Simultaneous (line ∩ parabola)
Set equal, form a quadratic; # solutions = # intersections = sign of its Δ\Delta.
Inequality (a>0)
ax2+bx+c<0ax^2+bx+c<0 holds between roots; >0>0 holds outside them.
Quadratic formula
x=−b±b2−4ac2ax = \tfrac{-b \pm \sqrt{b^2-4ac}}{2a}
Discriminant
Δ=b2−4ac\Delta = b^2 - 4ac
>0 two roots · =0 repeated · <0 none (real).
Vieta (quadratic)
α+β=−ba,    αβ=ca\alpha+\beta = -\tfrac{b}{a},\;\; \alpha\beta = \tfrac{c}{a}
Remainder / factor
Remainder of P(x)÷(x−a) is P(a); (x−a) is a factor ⟺ P(a)=0.
Common trap
Complex/irrational roots of real polynomials come in conjugate pairs.