Math Problem Statement

  1. Determine the value of k so that x 2 is a factor of x3-  2kx2 +6x - 4.

Solution

To determine the value of kk such that x2x - 2 is a factor of the polynomial x32kx2+6x4x^3 - 2kx^2 + 6x - 4, we need to use the Factor Theorem. According to the Factor Theorem, x2x - 2 is a factor of the polynomial if and only if the polynomial evaluates to zero when x=2x = 2.

Given the polynomial: f(x)=x32kx2+6x4f(x) = x^3 - 2kx^2 + 6x - 4

we need to find kk such that f(2)=0f(2) = 0.

Step-by-Step Solution:

  1. Substitute x=2x = 2 into the polynomial: f(2)=(2)32k(2)2+6(2)4f(2) = (2)^3 - 2k(2)^2 + 6(2) - 4

  2. Simplify the expression: f(2)=82k(4)+124f(2) = 8 - 2k(4) + 12 - 4 f(2)=88k+124f(2) = 8 - 8k + 12 - 4 f(2)=168kf(2) = 16 - 8k

  3. Set the expression equal to zero and solve for kk: 168k=016 - 8k = 0 8k=168k = 16 k=2k = 2

Conclusion:

The value of kk that makes x2x - 2 a factor of x32kx2+6x4x^3 - 2kx^2 + 6x - 4 is 2\boxed{2}.

Would you like any details or have any questions? Here are some possible follow-up questions:

  1. How does the Factor Theorem work?
  2. Can you show another example using the Factor Theorem?
  3. What if the polynomial had more than one unknown coefficient?
  4. How do you perform polynomial long division?
  5. Could you demonstrate synthetic division for this problem?

Tip: When using the Factor Theorem, always verify your solution by substituting the found value back into the polynomial to ensure it evaluates to zero.

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Math Problem Analysis

Mathematical Concepts

Polynomials
Factor Theorem

Formulas

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Theorems

Factor Theorem

Suitable Grade Level

Grades 10-12