Polynomials Theorems

Pre-Calc/Math Analysis BC (CHS)

Remainder Theorem

p(x) ÷ d(x) = q(x)d(x) +r
p(x) ÷ d(x) = q(x) + (r ÷ d(x))

Factor theorem

(x-k) is a factor of f(x) if f(k) = 0

rational root theorem

±b/±a if integer coefficients and has rational roots
±(constant’s factors) / ±(leading coefficient’s factors)

descarte’s rule of signs

max. pos. real roots: no. sign changes d’ f(x)

max neg. real roots: no. sign changes d’ f(-x)

intermediate value theorem

two points on positive-slope diagonal → must be some value between them vertical.m et horizontal.m

upper/lower bounds

synthetic division d’ f(x) by (x-c):

row 3 all nonnegative → c is upper bound d’ real roots

row 3 alternating nonnegative/nonpostivie → c is lower bound d’ real roots

conjugate pairs

if polynomial io real coefficients et root (a+bi), also has root (a-bi)

if polynomial io rational coefficients et root (a+b√c), also has root (a-b√c); index of root must be even

can have pair d’ +/- w/ odd index sed only coincidence, can w/o asw

Fundamental Theorem of Algebra

polynomial f(x) (single-variable, non-constant/non-zero): degree n → n complex roots

irreducible over rationals

neither irrationals nor imaginaries in factors

irreducible over reals

no imaginaries in factors

linearly factored

factored as much as possible with complex numbers

quadratic general form

f(x) = ax² + bx +c

quadratic vertex form

f(x) = a(x-h)² +k

quadratic factored form

f(x) = a(x-r₁)(x-r₂)

vertex (general)

(-b/2a, f(-b/2a) )

vertex (vertex form)

(h,k)

vertex (factored)

( (r₁+r₂)/2 , f(r₁+r₂)/2) )

axis of symmetry (general)

x = -b/2a

axis of symmetry (vertex)

x = h

axis of symmetry (factored)

x = (r₁ + r₂)/2

up

a > 0

down

a < 0

no real soln.’s

b²-4ac < 0

2 identical real zeroes

b²-4ac = 0

2 unique real zeroes

b²-4ac > 0