Math Problem Statement

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

Solution

To solve for cc, we use the fact that the polynomial 2x3+7x2+cx4=02x^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 rr be the double root. Then:

  1. P(r)=2r3+7r2+cr4=0P(r) = 2r^3 + 7r^2 + cr - 4 = 0 (root condition)
  2. P(r)=6r2+14r+c=0P'(r) = 6r^2 + 14r + c = 0 (derivative condition)

Step 2: Solve for cc using the conditions

From the derivative condition: c=6r214rc = -6r^2 - 14r Substitute this cc into the polynomial P(r)=0P(r) = 0: 2r3+7r2+(6r214r)r4=02r^3 + 7r^2 + (-6r^2 - 14r)r - 4 = 0 Simplify: 2r3+7r26r314r24=02r^3 + 7r^2 - 6r^3 - 14r^2 - 4 = 0 Combine like terms: 4r37r24=0-4r^3 - 7r^2 - 4 = 0 Factor: (4r3+7r2+4)=0- (4r^3 + 7r^2 + 4) = 0

Step 3: Solve 4r3+7r2+4=04r^3 + 7r^2 + 4 = 0

We solve this cubic equation by testing possible rational roots (using the Rational Root Theorem): The possible roots are ±1,±2,±4\pm 1, \pm 2, \pm 4.

Testing r=1r = -1: 4(1)3+7(1)2+4=4+7+4=7(not a root)4(-1)^3 + 7(-1)^2 + 4 = -4 + 7 + 4 = 7 \quad (\text{not a root}) Testing r=2r = -2: 4(2)3+7(2)2+4=32+28+4=04(-2)^3 + 7(-2)^2 + 4 = -32 + 28 + 4 = 0 So, r=2r = -2 is a root.

Step 4: Find cc

Substitute r=2r = -2 into the derivative condition: c=6(2)214(2)=6(4)+28=24+28=4c = -6(-2)^2 - 14(-2) = -6(4) + 28 = -24 + 28 = 4

Final Answer:

c=4c = 4


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

  1. How can you verify that r=2r = -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.

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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