Factor Theorem vs. Remainder Theorem
Factors and remainders play important roles in polynomial division and factorization. The factor theorem and remainder theorem are two key concepts that help in understanding the relationships between factors, remainders, and roots of polynomials.

Factor Theorem:
- The factor theorem states that if a polynomial \(f(x)\) has a factor \((x - a)\), then \(a\) is a root of the polynomial, meaning \(f(a) = 0\).
- In simpler terms, if you divide a polynomial by \((x - a)\) and the remainder is 0, then \((x - a)\) is a factor of the polynomial.
- The factor theorem is used to find roots or zeros of polynomials by testing different values of \(a\) to see if they make the polynomial equal to zero.

Remainder Theorem:
- The remainder theorem states that when a polynomial \(f(x)\) is divided by \((x - a)\), the remainder is \(f(a)\).
- In other words, if you divide a polynomial by \((x - a)\), the value of the polynomial at \(a\) will be the remainder.
- The remainder theorem is useful for finding the remainder when dividing polynomials and for evaluating polynomials at specific values without fully carrying out the division process.

Differences:
- The factor theorem helps in finding roots or zeros of polynomials, while the remainder theorem helps in finding remainders when dividing polynomials by linear factors.
- The factor theorem focuses on the relationship between factors and roots of polynomials, while the remainder theorem focuses on the relationship between divisors and remainders in polynomial division.
- The factor theorem is used to factorize polynomials by finding linear factors, while the remainder theorem is used to evaluate polynomials and find remainders efficiently.