**Factor Theorem**

**Theorem:** If p(x) is a polynomial of degree n ≥ 1 and a is any real number, then

(i) x – a is a factor of p(x), if p(a) = 0, and (ii) p(a) = 0, if x – a is a factor of p(x).

**Proof:** By the Remainder Theorem,

p(x) = (x – a) q(x) + p(a).

(i) If p(a) = 0, then p(x) = (x – a) q(x),

which shows that x – a is a factor of p(x).

(ii) Since x – a is a factor of p(x),

p(x) = (x – a) g(x) for same polynomial g(x).

In this case, p(a) = (a – a) g(a) = 0.

**To use factor theorem**

**Step 1:**(x + a) is factor of a polynomial p(x) if p(–a) = 0.**Step 2:**(ax – b) is a factor of a polynomial p(x) if p(b/a) = 0**Step 3:**ax + b is a factor of a polynomial p(x) if p(–b/a) = 0.**Step 4:**(x – a) (x – b) is a factor of a polynomial p(x) if p(a) = 0 and p(b) = 0.

**Factor Theorem Example Problems With Solutions**

**Example 1:** Examine whether x + 2 is a factor of x^{3} + 3x^{2} + 5x + 6 and of 2x + 4.

**Solution:** The zero of x + 2 is –2.

Let p(x) = x^{3} + 3x^{2} + 5x + 6 and s(x) = 2x + 4

Then, p(–2) = (–2)^{3} + 3(–2)^{2} + 5(–2) + 6

= –8 + 12 – 10 + 6

= 0

So, by the Factor Theorem, x + 2 is a factor of x^{3} + 3x^{2} + 5x + 6.

Again, s(–2) = 2(–2) + 4 = 0

So, x + 2 is a factor of 2x + 4.

**Example 2: **Use the factor theorem to determine whether x – 1 is a factor of

(a) x^{3} + 8x^{2} – 7x – 2

(b) 2x^{3} + 5x^{2} – 7

(c) 8x^{4} + 12x^{3} – 18x + 14

**Solution:**

**Example 3: **Factorize each of the following expression, given that x^{3} + 13 x^{2} + 32 x + 20. (x+2) is a factor.

**Solution:**

**Example 4: **Factorize x3 – 23 x^{2} + 142 x – 120

**Solution:**

**Example 5: **Show that (x – 3) is a factor of the polynomial x^{3} – 3x^{2} + 4x – 12

**Solution:**

**Example 6: **Show that (x – 1) is a factor of x^{10} – 1 and also of x^{11} – 1.

**Solution:**

**Example 7: **Show that x + 1 and 2x – 3 are factors of 2x^{3} – 9x^{2} + x + 12.

**Solution:**

**Example 8: **Find the value of k, if x + 3 is a factor of 3x^{2} + kx + 6.

**Solution: **

**Example 9: **If ax^{3} + bx^{2} + x – 6 has x + 2 as a factor and leaves a remainder 4 when divided by (x – 2), find the values of a and b.

**Solution:**

**Example 10: **If both x – 2 and x – 1/2 are factors of px^{2} + 5x + r, show that p = r.

**Solution:**

**Example 11: **If x^{2} – 1 is a factor of ax^{4} + bx^{3} + cx^{2} + dx + e, show that a + c + e = b + d = 0.

**Solution:**

**Example 12: **Using factor theorem, show that a – b, b – c and c – a are the factors of a(b^{2} – c^{2}) + b(c^{2} – a^{2}) + c(a^{2} – b^{2}).

**Solution:**