what is the difference between remainder and factor theorem?

the remainder theorem helps us find the reminder when a polynomial of degree n f(x)=a_nxⁿ+a_n-1x^n-1+.....+a₂x²+a₁x¹+a₀ (where a_n,....,a₀ are coefficients) is divided by another polynomial g(x)= ax+b of degree 1 ( a and b are here coefficients) the remainder can be found by setting g(x)=0 and finding the value of x and substituting it in the polynomial f(x) of degree n. in other words if a polynomial f(x) is divided by (ax+b) the remainder R=f(-b/a). example:let f(x)=x³ +2x²-3x+1 this is a polynomial of degree 3 as the highest power of x is 3. Let's consider that f(x) is to be divided by (2x-3) and we need to calculate the remainder which is f(3/2)=(3/2)³+2(3/2)²-3(3/2)+1=35/8
whereas, The factor theorem determines whether g(x) is a factor of f(x) (i.e R the remainder is 0 when f(x) is divided by g(x)) the factor theorem is very useful to determine roots of a polynomial. In other words if the remainder f(b/a)=0 then (ax-b) is a factor of f(x). example: (1) Is (x-5) a factor of f(x)=3x³-5x²-58x+40 first we need to check if the remainder is 0 when f(x) is divided by (x-5) and we can confirm that by checking that by calculating f(5)=3(5)³-5(5)²-58(5)+40=0 so indeed (x-5) is a factor
(2) determine if x=-3/2 is a root of f(x)=2x³+3x²-8x-12 we need to show that the remainder is 0 so we substitute in -3/2 f(-3/2)=0 so it is indeed a root and (2x+3) is a factor of f(x).