13.	When 𝑥³− 5𝑥²+ 6𝑥 − 12 is divided by 𝑥 − 4, the remainder is 

As mentioned earlier, to find the remainder when \( x^3 - 5x^2 + 6x - 12 \) is divided by \( x - 4 \), we apply the **Remainder Theorem**.

### Using the Remainder Theorem:
The Remainder Theorem states that the remainder when dividing a polynomial \( f(x) \) by \( x - c \) is equal to \( f(c) \).

For the given polynomial \( f(x) = x^3 - 5x^2 + 6x - 12 \) and the divisor \( x - 4 \), we substitute \( x = 4 \) into the polynomial to find the remainder:

### Step-by-Step Calculation:
\[
f(4) = 4^3 - 5(4^2) + 6(4) - 12
\]
\[
f(4) = 64 - 80 + 24 - 12
\]
\[
f(4) = -4
\]

### Final Answer:
The remainder when \( x^3 - 5x^2 + 6x - 12 \) is divided by \( x - 4 \) is **\(-4\)**.

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Do you need further clarification on this? Or would you like to explore related topics?

### Related Questions:
1. How do you perform synthetic division for \( x^3 - 5x^2 + 6x - 12 \) by \( x - 4 \)?
2. How is the Remainder Theorem applied to higher-degree polynomials?
3. What is the Factor Theorem and how does it relate to finding roots?
4. Can you use polynomial division to factor cubic equations?
5. What does it mean if the remainder is zero when dividing polynomials?
6. How can the Remainder Theorem help in finding rational roots?
7. What are some practical uses of the Remainder Theorem in calculus?
8. What are the steps for dividing polynomials using long division?

### Tip:
When dividing polynomials, using the Remainder Theorem is a quick way to find the remainder without doing the full division process!

Discover how to find the remainder when dividing the polynomial x^3 - 5x^2 + 6x - 12 by x - 4 using the Remainder Theorem. Detailed step-by-step solution provided.

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