Factor Theorem

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).

In conclusion, we have the following theorem: Theorem 5.3 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).

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).

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

∴ 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).

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).

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).

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

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

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).

Factorizing Polynomials by Factor Theorem

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.

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

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

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.

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

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.

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.

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).

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.

Factorize f(x) = 3x3 – 11x2 + 8x + 4. Step 1: List all the factors of leading coefficient 3 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.

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 = 4 0 f(–1) = 3(–1)3 – 11(–1)2 + 8(–1) + 4 = –18 0 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.

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)

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)

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

Follow-up question Factorize . By long division, ∴