1. Polynomials 5: The Factor theorem
Silent
Teacher
Intelligent Practice
Narration
Your Turn
Use the factor theorem to show that
𝑥−1 is a factor of 4 𝑥 3 −3 𝑥 2 −1.
Use the factor theorem to show that
𝑥−4 is a factor of −3 𝑥 𝑥 2 −6𝑥+8.
Use the factor theorem to show that
𝑥+1 is a factor of 𝑥 3 +3 𝑥 2 −33𝑥−35.
Practice

2. Use the factor theorem to show that
Worked Example
Your Turn
Use the factor theorem to show that
𝑥−2 is a factor of 𝑥 3 + 𝑥 2 −4𝑥−4.
Use the factor theorem to show that
𝑥+3 is a factor of 𝑥 3 + 2𝑥 2 −2𝑥+3.
Use the factor theorem to show that
𝑥−3 is NOT a factor of
𝑥 3 + 𝑥 2 −4𝑥−4.
Use the factor theorem to show that
𝑥+2 is NOT a factor of
𝑥 3 + 𝑥 2 +𝑥−3.

3. Use the factor theorem to show that 𝑥−1 is a factor of 3 𝑥 2 −5𝑥+2.
𝑥−2 is NOT a factor of 𝑥 3 −5 𝑥 2 −6𝑥+18.

4. Use the factor theorem to show that 𝑥−1 is a factor of 3 𝑥 2 −5𝑥+2.
If 𝑓 𝑥 =3 𝑥 2 −5𝑥+2, then f(1)=0. So (x-1) is a factor of 𝑓(𝑥).
Use the factor theorem to show that
𝑥−1 is a factor of 3 𝑥 3 +2 𝑥 2 −5.
If 𝑓 𝑥 =3 𝑥 3 +2 𝑥 2 −5, then f(1)=0. So (x-1) is a factor of 𝑓(𝑥).
If 𝑓 𝑥 =4 𝑥 3 −3 𝑥 2 −1, then f(1)=0. So (x-1) is a factor of 𝑓(𝑥).
Use the factor theorem to show that
𝑥−1 is a factor of 4 𝑥 3 −3 𝑥 2 −1.
Use the factor theorem to show that
𝑥−4 is a factor of −3 𝑥 𝑥 2 −6𝑥+8.
If 𝑓 𝑥 =−3 𝑥 𝑥 2 −6𝑥+8, then f(4)=0. So (x-4) is a factor of 𝑓(𝑥).
Use the factor theorem to show that
𝑥+1 is a factor of 𝑥 3 +3 𝑥 2 −33𝑥−35.
If 𝑓 𝑥 = 𝑥 3 +3 𝑥 2 −33𝑥−35, then f(-1)=0. So (x+1) is a factor of 𝑓(𝑥).
Use the factor theorem to show that
𝑥+3 is a factor of 5𝑥 4 −45 𝑥 2 −6𝑥−18.
If 𝑓 𝑥 = 5𝑥 4 −45 𝑥 2 −6𝑥−18, then f(-3)=0. So (x+3) is a factor of 𝑓(𝑥).
Use the factor theorem to show that
𝑥−2 is NOT a factor of 𝑥 3 −5 𝑥 2 −6𝑥+18.
If 𝑓 𝑥 = 𝑥 3 −5 𝑥 2 −6𝑥+18, then f(2)≠0. So (x-2) is NOT a factor of 𝑓(𝑥).