**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