Remainder and factor theorems
If we divide a polynomial by \((x − r)\), we obtain a result of the form:
$$f(x) = (x − r) q(x) + f(r)$$
where \(q(x)\) is a polynomial with one degree less than the degree of \(f(x)\) and \(f(r)\) is the remainder. This is called the remainder theorem.
If the remainder \(f(r) = 0\), then \((x − r)\) is a factor of \(f(x)\).
Is \((x-2)\) a factor of \(f(x)=x^3-2x-6\)?
We identify \(r\) as \(2\) and plug that value into our function:
$$ f(2) = 2^3 - 2 \cdot 2 + 6 = 8 - 4 + 6 = 10 $$
\((x-2)\) is not a factor of \(f(x) = x^3 - 2x - 6\)
Is \((x-1)\) a factor of \(f(x) = x^4 - 2x^2 + 1\)?