Revise: The Remainder Theorem and the Factor Theorem
Dividing polynomials by long division and synthetic division, and using the remainder and factor theorems to find and confirm roots without full division.
P(x) = D(x)·Q(x) + R(x), degree(R) < degree(D).
2x³−3x²−11x+6 ÷ (x−3) = 2x²+3x−2, remainder 0.
Dividing P(x) by (x−a) leaves remainder P(a).
P(x)=x³−2x²+4x−1, ÷(x−2) → remainder P(2)=7.
Test candidates, divide out each factor, repeat.
x³−2x²−5x+6 = (x−1)(x−3)(x+2).
Long division: divide, multiply, subtract, bring down — repeat.
x²+5x+6 ÷ (x+2) = x+3.
A fast coefficient-only shortcut for dividing by (x−a).
x³−4x²+x+6 ÷ (x−3) = x²−x−2.
Remainder = P(a), found by one substitution, no division needed.
P(x)=2x³+x−5, ÷(x+1) → remainder P(−1)=−8.
A direct consequence of the remainder theorem: remainder 0 ⇔ exact factor.
P(−2)=0 → (x+2) is a factor.
Test factors of the constant term, divide, repeat on the smaller quotient.
x³−7x+6 = (x−1)(x−2)(x+3).