Question

10. If the coefficient of x*4 is equal to that of x*6 in (1+x² + ax³)*4 what is the value of a

Answer 1

Answer:

a = 1/4

Explanation:

We can use the binomial theorem to expand (1 + x² + ax³)^4:

(1 + x² + ax³)^4 = (1)^4 + 4(1)^3(x²)(a) + 6(1)^2(x²)^2(a^2) + 4(1)(x²)^3(a^3) + (x²)^4(a^4)

Simplifying the terms with x^4 and x^6, we get:

x^4: 4a(x^2)^1 + (x^2)^4(a^4) = 4ax^2 + a^4x^8
x^6: 4(x^2)^2(a^2) + 4(x^2)^3(a^3) = 4a^2x^4 + 4a^3x^6

Since the coefficient of x^4 is equal to that of x^6, we can set their respective expressions equal to each other:

4ax^2 + a^4x^8 = 4a^2x^4 + 4a^3x^6

Dividing both sides by x^2, we get:

4a + a^4x^6 = 4a^2x^2 + 4a^3x^4

Since this must hold for all values of x, we can equate the coefficients of the powers of x on both sides:

For the coefficient of x^2: 4a = 4a^2
Simplifying, we get: a = 1/4

Therefore, the value of a is 1/4.

Answer 2

Answer: We can expand the given expression using the binomial theorem:

(1 + x² + ax³)^4 =

C(4,0) + C(4,1)x² + C(4,2)(ax³)^2 + C(4,3)x^6 + C(4,4)(ax³)^4

where C(n,k) denotes the binomial coefficient "n choose k", which is equal to n! / (k!(n-k)!).

The coefficient of x^4 is the coefficient of the second term, which is C(4,1) = 4. The coefficient of x^6 is the coefficient of the fourth term, which is C(4,3) = 4.

Since the coefficient of x^4 is equal to that of x^6, we have:

4 = C(4,1) = 4a C(4,3) = 4a

Solving for a, we get:

a = 1/4

Therefore, the value of a is 1/4.

Explanation: