Remainders, factors, roots and polynomial graphs.
Example 1
Find the remainder when P(x)=x3−2x+7 is divided by x−3.
Show worked answer
Answer: 28
- The remainder theorem gives remainder P(3).
- P(3)=27−6+7=28.
Example 2
x−2 is a factor of P(x)=x3+kx−14. Find k.
Show worked answer
Answer: 3
- The factor theorem requires P(2)=0.
- 8+2k−14=0.
- Hence 2k=6, so k=3.
Example 3
For y=2(x−3)2(x+1), what happens at x=3?
y
22.533.540510 x
y=2(x−3)2(x+1)Show worked answer
Answer: The graph touches the x−axis and turns
- The root x=3 has even multiplicity 2; the other factor is non-zero there.
- The polynomial keeps the same sign on both sides of this root, so its graph touches and turns.