factor theorem

in algebra, the factor theorem is a special case of the polynomial remainder theorem that is used to determine if an expression is a factor of a polynomial

if the remainder of $f(x)$ divided by $(ax+b)$, which is $f(\frac{-b}{a})$, equals zero, then $(ax+b)$ is a factor of $f$

applications of the factor theorem

• evaluating to see if $(ax+b)$ is a factor of $f(x)$
• finding coefficients of $f(x)$ when given the factor or remainder of $f(x)$ by $(ax+b)$

example

given the polynomial $f(x)=3x^2-4x+7$, determine if $(x-1)$ is a factor of $f$
according to the factor theorem, if the remainder of $f$ divided by $(x-1)$ is equal to $f(1)$ then $(x-1)$ is a factor of $f$
when we divide $f$ by $(x-1)$ we get $3x-1$ with a remainder of $6$, and $f(1)=6$ so $(x-1)$ is a factor of $f$

references

https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:poly-div/x2ec2f6f830c9fb89:remainder-theorem/v/polynomial-remainder-theorem