Use the Remainder and Factor Theorems to test for factors and find remainders when dividing polynomials.
The remainder theorem states that when a polynomial p(x) is divided by (x−a), the remainder is p(a) — found simply by substituting x=a.
Find the remainder when x³−4x+3 is divided by (x−2).
p(2)=8−8+3=3
Find the remainder when x²+3x−5 is divided by (x+1).
p(−1)=1−3−5=−7
The factor theorem states that (x−a) is a factor of p(x) if and only if p(a)=0.
Is (x−1) a factor of x³−6x²+11x−6?
p(1)=1−6+11−6=0, so yes, (x−1) is a factor
Is (x+2) a factor of x³+2x²−x−2?
p(−2)=−8+8+2−2=0, so yes, (x+2) is a factor
1. Find the remainder when x³−3x+1 is divided by (x−1).
2. Find the remainder when x²−5x+6 is divided by (x−3).
3. Is (x−2) a factor of x³−2x²−x+2?
4. Is (x+1) a factor of x³+3x²+3x+1?
5. Find k if (x−2) is a factor of x³−kx+6.