P(x)= 2x³-5x²+3x-7 using remainder theorem

To use the **Remainder Theorem**, we need to find the remainder when \( P(x) \) is divided by \( (x - a) \). According to the theorem, the remainder is simply \( P(a) \), i.e., the value of the polynomial when \( x = a \).

Let's say you want to find the remainder of \( P(x) = 2x^3 - 5x^2 + 3x - 7 \) when divided by \( (x - a) \). The remainder will be \( P(a) \), so you just need to substitute \( a \) into the polynomial.

### Example:

Let's use \( a = 2 \) as an example:

\[
P(2) = 2(2)^3 - 5(2)^2 + 3(2) - 7
\]

Calculating each term:

\[
P(2) = 2(8) - 5(4) + 3(2) - 7
\]
\[
P(2) = 16 - 20 + 6 - 7
\]
\[
P(2) = -5
\]

So, the remainder when dividing \( P(x) \) by \( (x - 2) \) is **-5**.

If you have a specific value of \( a \) in mind, I can substitute that value into the polynomial.

Would you like to explore any specific values of \( a \), or do you have further questions?

### Related Questions:
1. What is the remainder when \( P(x) = 2x^3 - 5x^2 + 3x - 7 \) is divided by \( x - 3 \)?
2. How is the remainder theorem used in synthetic division?
3. Can the remainder theorem be applied to polynomials with higher degrees (e.g., quartic or quintic)?
4. How does the remainder theorem help in finding the factors of a polynomial?
5. What is the relationship between the remainder theorem and the factor theorem?

### Tip:
To check if a number is a root of a polynomial, use the remainder theorem. If \( P(a) = 0 \), then \( x - a \) is a factor of the polynomial.

P(x) = 2x³ - 5x² + 3x - 7 using remainder theorem

Understand how to use the Remainder Theorem to find the remainder when the polynomial P(x) = 2x³ - 5x² + 3x - 7 is divided by (x - a). This solution explains each step of the process and provides a clear example. Ideal for students in Grades 9-12.

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