Math Problem Statement
2x^3+7x^2+cx-4=0. has a double root. c is an integer. find c
Solution
To solve for , we use the fact that the polynomial has a double root. This means that one root of the polynomial occurs twice, and it satisfies both the polynomial and its derivative.
Step 1: General conditions
Let be the double root. Then:
- (root condition)
- (derivative condition)
Step 2: Solve for using the conditions
From the derivative condition: Substitute this into the polynomial : Simplify: Combine like terms: Factor:
Step 3: Solve
We solve this cubic equation by testing possible rational roots (using the Rational Root Theorem): The possible roots are .
Testing : Testing : So, is a root.
Step 4: Find
Substitute into the derivative condition:
Final Answer:
Would you like a detailed check of this solution? Here are 5 follow-up questions to consider:
- How can you verify that is indeed a double root?
- What is the significance of the derivative condition in finding double roots?
- How does the Rational Root Theorem work in identifying potential roots?
- Can you generalize this approach for higher-degree polynomials?
- What happens if the polynomial has more than one double root?
Tip: Always test your solution by substituting back into the original equations to ensure accuracy.
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Math Problem Analysis
Mathematical Concepts
Algebra
Cubic Equations
Derivative of Polynomials
Roots of Polynomials
Rational Root Theorem
Formulas
P(r) = 2r^3 + 7r^2 + cr - 4 = 0
P'(r) = 6r^2 + 14r + c = 0
c = -6r^2 - 14r
Theorems
Rational Root Theorem
Suitable Grade Level
Grades 10-12
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