Find the value of k such that the coefficient of x^4 in the expansion of (x + k)^6 is 240.
Practice Questions
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Q1
Find the value of k such that the coefficient of x^4 in the expansion of (x + k)^6 is 240.
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The coefficient of x^4 is C(6,4) * k^2 = 15k^2. Setting 15k^2 = 240 gives k^2 = 16, so k = 4 or -4.
Questions & Step-by-step Solutions
1 item
Q
Q: Find the value of k such that the coefficient of x^4 in the expansion of (x + k)^6 is 240.
Solution: The coefficient of x^4 is C(6,4) * k^2 = 15k^2. Setting 15k^2 = 240 gives k^2 = 16, so k = 4 or -4.
Steps: 9
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.
