The product of two roots of the given equation is \[-10\].

\[10x^3 + 100x^2 = cx + 1000\]

**Find the third root.**

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# To find the third root, let's break the problem down step by step.

# Given:
# The equation is 10x^3 + 100x^2 = cx + 1000
# The product of two roots is -10.

# Let the roots be α, β, and γ.
# The equation can be expressed as:
# 10(x - α)(x - β)(x - γ) = 10x^3 + 100x^2 - cx - 1000

# We know the following:
# αβγ = -constant term / leading coefficient
# αβ + βγ + γα = -coefficient of x^2 / leading coefficient
# α + β + γ = -coefficient of x^2 / leading coefficient

# Let's calculate the product of all roots αβγ = -constant / leading coefficient

constant_term = -1000
leading_coefficient = 10

product_all_roots = -constant_term / leading_coefficient

# We know the product of two roots, let's say αβ = -10
product_two_roots = -10

# We can now find the third root using the relation αβγ = product_two_roots * γ
# γ = αβγ / αβ

third_root = product_all_roots / product_two_roots
third_rootThe third root of the given equation is \(\gamma = -10\).

Would you like further details or have any questions?

Here are 5 related questions to explore:

1. How would the equation change if the sum of the roots were provided instead?
2. What are the other two roots if one root is \(-10\)?
3. How would you find the value of \( c \) in the equation?
4. Can you derive the relationship between the coefficients and roots for any cubic equation?
5. How would the roots change if the constant term in the equation were different?

**Tip:** When solving cubic equations, identifying the relationships between the roots and coefficients can simplify the process significantly.

Learn how to find the third root of a cubic equation with the product of two roots given as -10. This problem involves polynomial equations, Vieta's formulas, and advanced algebraic concepts. Suitable for advanced high school students and those preparing for mathematical competitions.

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