Final answer:
To find the first three terms of the expression (1+kx)⁴ (1-4x)³, we need to expand the expression using the binomial theorem. The first three terms are 1, (4k-12k)x, and (6k²-48k²+12k)x². To find the coefficient of x², we set the expression (6k²-48k²+12k) equal to zero and solve for k. The value of k is 1/3.
Step-by-step explanation:
To find the first three terms of the expression (1+kx)⁴ (1-4x)³, we need to expand the expression using the binomial theorem.
First, let's expand (1+kx)⁴ using the binomial theorem:
(1+kx)⁴ = 1 + 4kx + 6k²x² + 4k³x³ + k⁴
Now, let's expand (1-4x)³ using the binomial theorem:
(1-4x)³ = 1 - 3(4x) + 3(4x)² - (4x)³
Multiplying the two expanded expressions:
(1+kx)⁴ (1-4x)³ = (1 + 4kx + 6k²x² + 4k³x³ + k⁴)(1 - 3(4x) + 3(4x)² - (4x)³)
Expanding further:
(1+kx)⁴ (1-4x)³ = 1 + 4kx + 6k²x² + 4k³x³ + k⁴ - 12kx - 48k²x² - 48k³x³ + 12kx² + 48k²x³ - 12k³x⁴
Combining like terms:
(1+kx)⁴ (1-4x)³ = 1 + (4k-12k)x + (6k²-48k²+12k)x² + (4k³-48k³)x³ - 12k³x⁴
The first three terms of the expanded expression are:
1, (4k-12k)x, (6k²-48k²+12k)x²
To find the coefficient of x², we need to look at the expression (6k²-48k²+12k). Since the coefficient of x² is zero, we can set this expression equal to zero and solve for k:
6k²-48k²+12k = 0
Combining like terms:
-42k² + 12k = 0
Factoring out k:
k(-42k + 12) = 0
Setting each factor equal to zero:
k = 0 or -42k + 12 = 0
If k = 0, the coefficient of x² is also zero.
Solving -42k + 12 = 0 for k:
-42k = -12
k = -12/-42
k = 1/3
If k = 1/3, the coefficient of x² is also zero.