Find the value of k such that the coefficient of x^4 in the expansion of (x + k)

Find the value of k such that the coefficient of x^4 in the expansion of (x + k)^6 is 240.

Find the value of k such that the coefficient of x^4 in the expansion of (x + k)^6 is 240.

Correct Answer: 4 or -4

Step 1: Understand that we need to find the coefficient of x^4 in the expansion of (x + k)^6.
Step 2: Use the binomial theorem, which tells us how to expand (a + b)^n. The coefficient of x^4 in (x + k)^6 is given by the formula C(6, 4) * k^(6-4).
Step 3: Calculate C(6, 4), which is the number of ways to choose 4 items from 6. This is equal to 15.
Step 4: The expression for the coefficient of x^4 becomes 15 * k^2.
Step 5: We know from the problem that this coefficient equals 240, so we set up the equation: 15k^2 = 240.
Step 6: Solve for k^2 by dividing both sides of the equation by 15: k^2 = 240 / 15.
Step 7: Calculate 240 / 15, which equals 16, so now we have k^2 = 16.
Step 8: To find k, take the square root of both sides: k = ±4.
Step 9: Therefore, the possible values for k are 4 and -4.

Binomial Expansion – Understanding how to find coefficients in the expansion of a binomial expression using the binomial theorem.
Combinatorics – Applying combinations to determine the coefficients in the expansion.
Quadratic Equations – Solving for k involves recognizing and solving a quadratic equation.