Use the remainder theorem to find P(-1) for $$ P\left(x\right)=x^3+4x^2+7 $$

To find $ P(-1) $ for the polynomial $ P(x) = x^3 + 4x^2 + 7 $ using the Remainder Theorem, we follow these steps:

The Remainder Theorem states that if a polynomial $ P(x) $ is divided by $ x - a $, the remainder of this division is $ P(a) $. Thus, to find $ P(-1) $, we can simply substitute $ x = -1 $ into the polynomial.

Given:
\[ P(x) = x^3 + 4x^2 + 7 \]

Substitute $ x = -1 $:
\[ P(-1) = (-1)^3 + 4(-1)^2 + 7 \]

Now, calculate each term:
\[ (-1)^3 = -1 \]
\[ 4(-1)^2 = 4 \cdot 1 = 4 \]
\[ 7 \text{ remains as it is} \]

Combine the results:
\[ P(-1) = -1 + 4 + 7 \]
\[ P(-1) = 10 \]

Therefore, $ P(-1) = 10 $.

Would you like any 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 the Remainder Theorem to determine the roots of a polynomial?
3. What is the value of $ P(2) $ for the same polynomial?
4. How would you apply the Factor Theorem to $ P(x) = x^3 + 4x^2 + 7 $?
5. What are synthetic division and how does it relate to the Remainder Theorem?
6. How do you factorize a cubic polynomial?
7. What are the roots of $ P(x) = x^3 + 4x^2 + 7 $?
8. How do you graph the polynomial $ P(x) = x^3 + 4x^2 + 7 $?

**Tip:** Always check if you can factorize a polynomial to simplify the evaluation process.

Learn how to use the Remainder Theorem to find P(-1) for the polynomial P(x) = x^3 + 4x^2 + 7. Step-by-step solution provided, including calculation of each term and final result. Suitable for high school students and anyone studying polynomials and algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation