Without dividing out, find the remainder when 2x 2 −3x + 4 is divided by (x −2) by the remainder theorem

To find the remainder when the polynomial \( f(x) = 2x^2 - 3x + 4 \) is divided by \( x - 2 \), we can use the **Remainder Theorem**. The theorem states that if a polynomial \( f(x) \) is divided by \( x - c \), the remainder is \( f(c) \).

Here, \( c = 2 \), so we need to find \( f(2) \):

\[
f(x) = 2x^2 - 3x + 4
\]

Substitute \( x = 2 \) into \( f(x) \):

\[
f(2) = 2(2)^2 - 3(2) + 4
\]

\[
f(2) = 2(4) - 3(2) + 4
\]

\[
f(2) = 8 - 6 + 4
\]

\[
f(2) = 6
\]

So, the remainder when \( 2x^2 - 3x + 4 \) is divided by \( x - 2 \) is **6**.

Would you like a more detailed explanation, or do you have any questions about this problem?

Here are some related questions you might find interesting:
1. What is the remainder theorem, and why does it work?
2. How would you find the remainder if the divisor was \( x + 3 \)?
3. Can the remainder theorem be used with divisors other than linear polynomials?
4. How does the factor theorem relate to the remainder theorem?
5. What is synthetic division, and how does it relate to finding remainders?
6. What happens if \( f(c) = 0 \) when using the remainder theorem?
7. Can the remainder theorem be applied to higher degree polynomials?
8. How do you interpret the remainder in the context of polynomial division?

**Tip:** When working with polynomial division, always remember to check if the divisor can be factored further, as it can sometimes simplify the division process.

Learn how to find the remainder when dividing the polynomial 2x^2 - 3x + 4 by (x - 2) using the Remainder Theorem. Follow a step-by-step explanation with detailed calculations and understand the application of the theorem in polynomial division. Suitable for high school students studying algebra.

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