1. Let's start by stating the problem: You want to understand the Factor Theorem and the Remainder Theorem, which are important tools in algebra for working with polynomials. 2. The Remainder Theorem states that if a polynomial $f(x)$ is divided by a linear divisor of the form $x - a$, then the remainder of this division is $f(a)$. 3. This means to find the remainder when dividing $f(x)$ by $x - a$, you simply evaluate the polynomial at $x = a$. 4. The Factor Theorem is a special case of the Remainder Theorem. It states that $x - a$ is a factor of the polynomial $f(x)$ if and only if $f(a) = 0$. 5. In other words, if substituting $a$ into the polynomial gives zero, then $x - a$ divides the polynomial exactly with no remainder. 6. To apply these theorems, follow these steps: - Given a polynomial $f(x)$ and a value $a$, calculate $f(a)$. - If $f(a) = 0$, then $x - a$ is a factor of $f(x)$ (Factor Theorem). - If $f(a) \neq 0$, then $f(a)$ is the remainder when $f(x)$ is divided by $x - a$ (Remainder Theorem). 7. Example: Let $f(x) = x^3 - 4x^2 + x + 6$ and check if $x - 2$ is a factor. - Calculate $f(2) = 2^3 - 4(2)^2 + 2 + 6 = 8 - 16 + 2 + 6 = 0$. - Since $f(2) = 0$, $x - 2$ is a factor of $f(x)$. 8. These theorems help simplify polynomial division and factorization, making it easier to solve polynomial equations. Final answer: The Factor Theorem tells us that $x - a$ is a factor of $f(x)$ if $f(a) = 0$, and the Remainder Theorem tells us the remainder when dividing $f(x)$ by $x - a$ is $f(a)$.