1. Factor Theorem

2. Consider a polynomial f(x) divided by x – a.
By the remainder theorem, we have
remainder = f(a)
If remainder = 0, then f(x) is divisible by x – a.
i.e.
If f(a) = 0, then x – a is a factor of f(x).

3. In conclusion, we have the following theorem:
Theorem Factor theorem
For a polynomial f(x), if f(a) = 0, then x – a is a factor of f(x).
f(a) = 0
x – a is a factor of f(x).
Consider f(x) = x2 – 5x + 6.
f(2) = (2)2 – 5(2) + 6 = 0
∴ x – 2 is a factor of f(x).

4. Theorem 5.4 | ø ö ç è æ | ø ö ç è æ How about if a is not an integer?
The factor theorem can be generalized as follows:
Theorem 5.4
For a polynomial f(x), if f = 0, then mx – n is a
factor of f(x). | ø ö ç è æ m n
f = 0 | ø ö ç è æ m n
mx – n is a factor of f(x).

5. Is 2x – 1 a factor of 2x2 + x – 1?
Let f(x) = 2x2 + x – 1.
∴ 2x – 1 is a factor of 2x2 + x – 1.

6. ∴ x – 2 is not a factor of f(x).
Follow-up question
Let f(x) = 2x2 – 5x – 3. Use factor theorem to determine whether each of the following is a factor of f(x).
(a) x – 2 (b) 2x + 1
(a) f(2) = 2(2)2 – 5(2) – 3
= 8 – 10 – 3
= –5
 0
∴ x – 2 is not a factor of f(x).

7. Follow-up question
Let f(x) = 2x2 – 5x – 3. Use factor theorem to determine whether each of the following is a factor of f(x).
(a) x – 2 (b) 2x + 1
(b)
∴ 2x + 1 is a factor of f(x).

8. In fact, the converse of the factor theorem is also true.
Consider a polynomial f(x).
If x – a is a factor of f(x), then f(a) = 0.
(a)
f(a) = 0
x – a is a factor of f(x).
(b) If mx – n is a factor of f(x), then f = 0. | ø ö ç è æ m n
f = 0 | ø ö ç è æ m n
mx – n is a factor of f(x).

9. If x + 3 is a factor of f(x) = –4x2 + px + 6, what is the value of p?
 By the converse of the factor theorem
–4(–3)2 + p(–3) + 6 = 0
–36 – 3p + 6 = 0
–3p = 30
p = –10

10. Follow-up question
If 8x3 + 6x2 + qx + 3 is divisible by 4x – 3, find the value of q.
Let f(x) = 8x3 + 6x2 + qx + 3.
i.e. 4x – 3 is a factor of f(x).
∵ f(x) is divisible by 4x – 3, 4 3 = | ø ö ç è æ f ∴ 3 4 6 8 2 = + | ø ö ç è æ q 4 39 3 - = q 13 - = q

11. Consider a polynomial f(x). It is known that f(a) = 0.
When f(x) is divided by x – a,
remainder = f(a)
= 0
by division algorithm, we have
f(x)  (x – a)Q(x) + 0
 (x – a)Q(x)
 x – a is a factor of f(x).

12. Factorizing Polynomials by Factor Theorem

13. How can we factorize a cubic polynomial?
The key step is to find a linear factor of a polynomial by the factor theorem. Let’s consider
f(x) = 2x3 + x2 – 4x – 3.

14. Find a linear factor of f(x) = 2x3 + x2 – 4x – 3.
Suppose f(x) can be factorized as (mx + n)(ax2 + bx + c),
i.e. 2x3 + x2 – 4x – 3  (mx + n)(ax2 + bx + c), …… (*)
where m, n, a, b and c are integers with m > 0.
Consider the coefficient of x3.
2x3 + x2 – 4x – 3  (mx + n)(ax2 + bx + c)
2x3
max3
By comparing the coefficient of x3 on both sides, we have
2 = ma

15. Find a linear factor of f(x) = 2x3 + x2 – 4x – 3.
Suppose f(x) can be factorized as (mx + n)(ax2 + bx + c),
i.e. 2x3 + x2 – 4x – 3  (mx + n)(ax2 + bx + c), …… (*)
where m, n, a, b and c are integers with m > 0.
Consider the constant term.
2x3 + x2 – 4x – 3  (mx + n)(ax2 + bx + c) –3 nc
By comparing the constant term on both sides, we have
–3 = nc

16. Find a linear factor of f(x) = 2x3 + x2 – 4x – 3.
Suppose f(x) can be factorized as (mx + n)(ax2 + bx + c),
i.e. 2x3 + x2 – 4x – 3  (mx + n)(ax2 + bx + c), …… (*)
where m, n, a, b and c are integers with m > 0.
2 = ma
Possible values of m = 1 and 2
–3 = nc
Possible values of n = ± 1 and ± 3 ∴
The possible linear factors of f(x) are
x ± 1, x ± 3, 2x ± 1, 2x ± 3.

17. Then, apply the factor theorem to the possible linear factors.
∴ x + 1 is a factor of f(x).

18. Consider a cubic polynomial f(x) = px3 + qx2 + rx + s, where p, q, r and s are integers. If mx + n is a linear factor of f(x), then we have:
px3 + qx2 + rx + s  (mx + n) Q(x), where Q(x) is a polynomial.
We can see that m is a factor of leading coefficient p, and n is a factor of the constant term s.
Therefore, a linear factor mx + n can be found as follows:
1. List all the factors of leading coefficient p and constant term s.
2. Apply the factor theorem to possible linear factors until a linear factor mx + n is found.

19. Follow-up question Consider f(x) = 3x3 – 11x2 – 19x – 5. (a)
By considering the coefficient of x3 and the constant term, which of the following is/are possible linear factor(s) of f(x)?
x + 1, 2x – 1, 3x + 2, 3x – 5
(b)
Among the result(s) obtained in (a), which of them is a factor of f(x)?
(a)
∵ Leading coefficient = 3 and constant term = –5
∴ Possible values of m = 1 and 3,
possible values of n = ± 1 and ± 5.
∴ The possible linear factors of f(x) are x + 1 and 3x – 5.

20. Follow-up question Consider f(x) = 3x3 – 11x2 – 19x – 5. (a)
By considering the coefficient of x3 and the constant term, which of the following is/are possible linear factor(s) of f(x)?
x + 1, 2x – 1, 3x + 2, 3x – 5
(b)
Among the result(s) obtained in (a), which of them is a factor of f(x)?
(b)
f(–1) = 3(–1)3 – 11(–1)2 – 19(–1) – 5 = 0
∴ x + 1 is a factor of f(x).
∴ 3x – 5 is not a factor of f(x).

21. Let consider f(x) = 3x3 – 11x2 + 8x + 4
Let consider f(x) = 3x3 – 11x2 + 8x + 4. We can summarize the steps for factorizing a cubic polynomial.

22. Factorize f(x) = 3x3 – 11x2 + 8x + 4.
Step 1: List all the factors of leading coefficient and constant term +4.
Positive factor of the leading coefficient 3:
1, 3
Factors of the constant term +4:
±1, ±2, ±4
 Possible linear factors of f(x):
x ± 1, x ± 2, x ± 4, 3x ± 1 , 3x ± 2, 3x ± 4.

23. Factorize f(x) = 3x3 – 11x2 + 8x + 4.
Step 2: Use the factor theorem to find a factor of f(x).
f(1) = 3(1)3 – 11(1)2 + 8(1) + 4 =
f(–1) = 3(–1)3 – 11(–1)2 + 8(–1) + 4 = –
f(2) = 3(2)3 – 11(2)2 + 8(2) + 4 = 0
 x – 2 is a factor of f(x).
After we find a factor of f(x), we can perform long division.

24. Factorize f(x) = 3x3 – 11x2 + 8x + 4.
Step 3: Perform long division. 4 8 11 2 3 + - x 3x 2 4 8 3 - + 5x 3x x
10x
-5x 6x
the quotient of f(x) ÷ (x – 2).
a factor of f(x) 4 +
 f(x) = (x – 2)(3x2 – 5x – 2)

25. Factorize f(x) = 3x3 – 11x2 + 8x + 4.
Step 4: Further factorize the quotient if possible.
f(x) = (x – 2)(3x2 – 5x – 2)
= (x – 2)(x – 2)(3x + 1)
Factorize 3x2 – 5x – 2.
Rough Work
3x +1
x –2
+x –6x = –5x
∴ f(x) = (x – 2)2(3x + 1)

26. Follow-up question
Factorize . ∵
∴ x – 3 is a factor of f(x).

27. Follow-up question
Factorize .
By long division, ∴