Question

Explain factor theorem?

Answer 1

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.
For Example:
Examine whether x + 2 is a factor of $x^3+3x^2+5x+6$
The zero of x + 2 is -2. Let $p\left(x\right)=x^3+3x^2+5x+6$ and $s\left(x\right)=2x+4$
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 $x^3+3x^2+5x+6$