use the remainder theorem to find P(-3) for $$ P\left(x\right)=-x^3-3x^2-x+6 $$

The Remainder Theorem states that for any polynomial $ P(x) $, if you divide $ P(x) $ by $ x - c $, the remainder of that division is $ P(c) $. To find $ P(-3) $ for the polynomial $ P(x) = -x^3 - 3x^2 - x + 6 $, we simply evaluate the polynomial at $ x = -3 $.

Let's calculate $ P(-3) $:

\[ P(x) = -x^3 - 3x^2 - x + 6 \]

\[ P(-3) = -(-3)^3 - 3(-3)^2 - (-3) + 6 \]

First, calculate each term separately:

1. $(-3)^3 = -27$
2. $ -(-27) = 27 $
3. $(-3)^2 = 9$
4. $ -3 \times 9 = -27 $
5. $-(-3) = 3 $
6. The constant term remains $6$

Now, sum these values:

\[ P(-3) = 27 - 27 + 3 + 6 \]
\[ P(-3) = 27 - 27 + 9 \]
\[ P(-3) = 9 \]

Thus, $ P(-3) = 9 $.

Would you like more details or have any questions?

Here are some related questions for further exploration:

1. What is the Remainder Theorem?
2. How can you use synthetic division to find $ P(-3) $?
3. What is the factor theorem and how is it related to the remainder theorem?
4. Can the Remainder Theorem be used for polynomials with complex coefficients?
5. How do you determine the roots of a polynomial using the Remainder Theorem?
6. What is the process for dividing polynomials?
7. How does the Remainder Theorem help in polynomial interpolation?
8. How can you verify your result from the Remainder Theorem using a graphing calculator?

**Tip:** When evaluating polynomials, always simplify each term separately before combining them to avoid mistakes.

Learn how to use the Remainder Theorem to find P(-3) for the polynomial P(x) = -x^3 - 3x^2 - x + 6 with a step-by-step solution. This problem explores polynomials and the application of the Remainder Theorem in algebraic computations. Suitable for high school students and above.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation