https://file.aiquickdraw.com/mathbot/file/1726451561191hebhpuux.jpg

The given problem states that \( x + 2 \) is a factor of the polynomial:

\[
x^4 - 10x^3 + 6x^2 + 10x - 100
\]

We can apply the **factor theorem** to verify this. The factor theorem states that if \( x + 2 \) is a factor of the polynomial, then \( f(-2) = 0 \), where \( f(x) \) is the given polynomial.

Let's compute \( f(-2) \):

\[
f(x) = x^4 - 10x^3 + 6x^2 + 10x - 100
\]

Substituting \( x = -2 \):

\[
f(-2) = (-2)^4 - 10(-2)^3 + 6(-2)^2 + 10(-2) - 100
\]
\[
= 16 - 10(-8) + 6(4) + 10(-2) - 100
\]
\[
= 16 + 80 + 24 - 20 - 100
\]
\[
= 100 - 20 - 100 = -20
\]

Since \( f(-2) = -20 \), this shows that \( x + 2 \) is **not a factor** of the polynomial \( x^4 - 10x^3 + 6x^2 + 10x - 100 \).

Would you like more details or have any further questions?

Here are five related questions for further exploration:
1. What does it mean for a polynomial to have a specific factor?
2. How can synthetic division be used to verify if \( x + 2 \) is a factor?
3. Can the given polynomial be factored using other methods?
4. What is the significance of the factor theorem in solving polynomial equations?
5. How can we factor polynomials of higher degrees systematically?

**Tip**: When testing whether a binomial is a factor of a polynomial, the factor theorem is a quick way to verify it without going through the entire division process!

x + 2 is a factor of x^4 - 10x^3 + 6x^2 + 10x - 100. Is this true?

Explore how to determine whether x + 2 is a factor of the polynomial x^4 - 10x^3 + 6x^2 + 10x - 100 using the Factor Theorem. This detailed guide explains step-by-step calculations and highlights key algebraic concepts, ideal for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation