REVIEW OF CHAPTER: THEORY OF EQUATIONS

P(x) =

long division
synthetic division x-r
example 1
x + 3
x-(-3)
r = -3
example 2
x-3
r=3

1) P(x) = ax + b r = 0 1 answer & 1 factor
   P(x) = a(x-r)
        P(x) = 3x - 4 P(x) = 3(x-)
        3x - 4 = 0
            3x = 4
            x =

2) P(x) = ax2 + bx + c
            solving equations roots (ans)
            and finding zeros of P(x) zeros (ans)

a.  3 methods

1. completing the square
2. factoring
3. quad form

b.  sum of 2 zeros r1 + r2 =
    product of 2 zeros r1 r2 =

c. discriminant
    b2 - 4ac > 0 real and unequal
    b2 - 4ac= 0 real and equal
    b2 - 4ac< 0 complex conj.

d. expressions quadratic in form
    x4 + 5x2 + 6 = 0 4th deg = 4 ans.
    let x2 = m

e. radical equations and also equations to fractional powers

    = x-2 x2/3 + 3x1/3 + 4 = 0
    let m = x 1/3
    MUST CHECK ALL ANSWERS

f.  P(x) = ax2 + bx + c
    P(x) = a(x - r1)(x - r2) 2 factors

3)  General P(x) = a(x - r1)(x - r2)(x - r3)(x - rn)

    nth degree - "n" zeros = "n" linear factors

a. Remainder Theorem - If P(x) is divided by x - r and R is the remainder then P(r ) = R

b. Factor Theorem - If x - r is a factor of P(x), then R = 0 (So P® = 0 if x - r is a factor)

c. P(x) with real coefficients (no i's) If r is a zero, so is r (really, only pertains to complex #'s)

d. P(x) with integral coefficients (whole #'s) If rational zeros exist, they are in the form of p/q, where p is a factor of a0(constant) and q is a factor of an(leading coefficient)