2x^3+7x^2+cx-4=0. has a double root. c is an integer. find c

To solve for \( c \), we use the fact that the polynomial \( 2x^3 + 7x^2 + cx - 4 = 0 \) 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 \( r \) be the double root. Then:
1. \( P(r) = 2r^3 + 7r^2 + cr - 4 = 0 \)  (root condition)
2. \( P'(r) = 6r^2 + 14r + c = 0 \) (derivative condition)

### Step 2: Solve for \( c \) using the conditions
From the derivative condition:
\[
c = -6r^2 - 14r
\]
Substitute this \( c \) into the polynomial \( P(r) = 0 \):
\[
2r^3 + 7r^2 + (-6r^2 - 14r)r - 4 = 0
\]
Simplify:
\[
2r^3 + 7r^2 - 6r^3 - 14r^2 - 4 = 0
\]
Combine like terms:
\[
-4r^3 - 7r^2 - 4 = 0
\]
Factor:
\[
- (4r^3 + 7r^2 + 4) = 0
\]

### Step 3: Solve \( 4r^3 + 7r^2 + 4 = 0 \)
We solve this cubic equation by testing possible rational roots (using the Rational Root Theorem):
The possible roots are \( \pm 1, \pm 2, \pm 4 \).

Testing \( r = -1 \):
\[
4(-1)^3 + 7(-1)^2 + 4 = -4 + 7 + 4 = 7 \quad (\text{not a root})
\]
Testing \( r = -2 \):
\[
4(-2)^3 + 7(-2)^2 + 4 = -32 + 28 + 4 = 0
\]
So, \( r = -2 \) is a root.

### Step 4: Find \( c \)
Substitute \( r = -2 \) into the derivative condition:
\[
c = -6(-2)^2 - 14(-2) = -6(4) + 28 = -24 + 28 = 4
\]

### Final Answer:
\[
c = 4
\]

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Would you like a detailed check of this solution? Here are 5 follow-up questions to consider:

1. How can you verify that \( r = -2 \) is indeed a double root?
2. What is the significance of the derivative condition in finding double roots?
3. How does the Rational Root Theorem work in identifying potential roots?
4. Can you generalize this approach for higher-degree polynomials?
5. 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.

Learn how to solve for the integer value of c in the polynomial 2x^3 + 7x^2 + cx - 4, given that it has a double root. This problem involves key algebraic concepts such as cubic equations, the derivative of polynomials, and the Rational Root Theorem. Ideal for high school students studying algebra and polynomials.

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