← Polynomials & Complex Numbers
Mathematics · Topic Cheatsheet

Polynomials & Complex Numbers

30 key results accumulated across 3 chapters.

Standard form
Ch 1
y=ax2+bx+c(a≠0)y = ax^2 + bx + c \quad (a \neq 0)
Vertex form
Ch 1
y=a(x−h)2+ky = a(x-h)^2 + k
Vertex at (h,k)(h, k).
Factored form
Ch 1
y=a(x−p)(x−q)y = a(x-p)(x-q)
Roots at x=p,qx = p, q.
Axis of symmetry / vertex x
Ch 1
x=−b2ax = -\frac{b}{2a}
Quadratic formula
Ch 1
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
Discriminant
Ch 1
Δ=b2−4ac\Delta = b^2 - 4ac
>0>0: 2 roots · =0=0: 1 repeated · <0<0: none real.
Simultaneous (line ∩ parabola)
Ch 1
Set equal, form a quadratic; # solutions = # intersections = sign of its Δ\Delta.
Inequality (a>0)
Ch 1
ax2+bx+c<0ax^2+bx+c<0 holds between roots; >0>0 holds outside them.
Quadratic formula
Ch 1
x=−b±b2−4ac2ax = \tfrac{-b \pm \sqrt{b^2-4ac}}{2a}
Discriminant
Ch 1
Δ=b2−4ac\Delta = b^2 - 4ac
>0 two roots · =0 repeated · <0 none (real).
Vieta (quadratic)
Ch 1
α+β=−ba,    αβ=ca\alpha+\beta = -\tfrac{b}{a},\;\; \alpha\beta = \tfrac{c}{a}
Remainder / factor
Ch 1
Remainder of P(x)÷(x−a) is P(a); (x−a) is a factor ⟺ P(a)=0.
Common trap
Ch 1
Complex/irrational roots of real polynomials come in conjugate pairs.
Imaginary unit
Ch 2
i=−1,i2=−1i = \sqrt{-1},\quad i^2 = -1
Complex number
Ch 2
z=a+biz = a + bi
real part aa, imaginary part bb.
Conjugate
Ch 2
z‾=a−bi,zz‾=a2+b2\overline{z} = a - bi,\quad z\overline{z} = a^2 + b^2
Polar form
Ch 2
z=r(cos⁡θ+isin⁡θ)=r cis θz = r(\cos\theta + i\sin\theta) = r\,\text{cis}\,\theta
Modulus / argument
Ch 2
r=a2+b2,θ=arg⁡(z)r = \sqrt{a^2+b^2},\quad \theta = \arg(z)
De Moivre
Ch 2
[r cis θ]n=rn cis(nθ)[r\,\text{cis}\,\theta]^n = r^n\,\text{cis}(n\theta)
Forms
Ch 2
z=a+bi=r(cos⁡θ+isin⁡θ)=reiθz = a+bi = r(\cos\theta + i\sin\theta) = re^{i\theta}
r = |z| = √(a²+b²), θ = arg z.
Modulus / conjugate
Ch 2
∣z∣2=zzˉ,    a+bi‾=a−bi|z|^2 = z\bar z,\;\; \overline{a+bi}=a-bi
De Moivre
Ch 2
zn=rn(cos⁡nθ+isin⁡nθ)z^n = r^n(\cos n\theta + i\sin n\theta)
Common trap
Ch 2
arg is measured from +x axis; watch the quadrant (add/subtract π).
Methods
Ch 3
Substitution or elimination; reduce 3×3 to 2×2.
Unique solution
Ch 3
Lines/planes meet at one point.
No solution
Ch 3
Parallel / inconsistent.
Infinite solutions
Ch 3
Same line/plane (dependent equations).
Methods
Ch 3
Substitution, elimination, or matrices (row reduction). 3 unknowns ⇒ 3 independent equations.
Number of solutions
Ch 3
Unique (lines meet), none (parallel/inconsistent), or infinite (same line/plane).
Common trap
Ch 3
Always check a solution in ALL original equations.