This question covers polynomial division, the Remainder Theorem, and the Factor Theorem.
When a polynomial P(x) of degree ≤4 is divided by a divisor D(x), the result satisfies:
P(x)=D(x)⋅Q(x)+R(x)
where Q(x) is the quotient and R(x) is the remainder, with deg(R)<deg(D).
Synthetic Division (for linear divisor x−k):
- Write the coefficients of P(x) in order (use 0 for missing terms).
- Bring down the leading coefficient.
- Multiply by k, add to next coefficient; repeat.
- The last value is the remainder; the others form the quotient coefficients.
Remainder Theorem: When P(x) is divided by (x−a), the remainder equals P(a).
Example: Find the remainder when P(x)=2x3+3x2−4x+1 is divided by (x+2).
Here a=−2, so:
P(−2)=2(−2)3+3(−2)2−4(−2)+1=−16+12+8+1=5
The remainder is 5.
Factor Theorem: (x−a) is a factor of P(x) if and only if P(a)=0.
This is a direct consequence of the Remainder Theorem — if the remainder is zero, the divisor is a factor.
Steps to factorize a cubic polynomial P(x):
- Find a root by testing integer values a (factors of the constant term) until P(a)=0.
- Divide P(x) by (x−a) using synthetic or long division to get a quadratic Q(x).
- Factorize Q(x) using the quadratic formula or inspection.
- Write P(x)=(x−a)⋅Q(x) fully factored.
Example: Factorize P(x)=x3−6x2+11x−6.
Test x=1: P(1)=1−6+11−6=0 ✓
Divide by (x−1):
P(x)=(x−1)(x2−5x+6)=(x−1)(x−2)(x−3)