Question

If \( f(x) = ax + bx^2 \), then the coefficient of \( x^3 \) in \( f(f(x)) \) is

• A. \( 0 \)
• B. \( ab \)
• C. \( a^3 \)
• D. \( ab^2 \)
• E. \( 2ab^2 \)

Hint:

Whenever you need the coefficient of a particular power in a composite function, first write the composition carefully, then expand only up to the required power. This saves time and reduces mistakes.

Answer 1

Answer: (E)

Step 1: Write the given function clearly.
We are given \[ f(x)=ax+bx^2 \] and we need to find the coefficient of \(x^3\) in \(f(f(x))\).
This means we have to substitute \(f(x)\) in place of \(x\) in the same function.

Step 2: Use the definition of composition of functions.
Since \[ f(t)=at+bt^2 \] for any input \(t\), replacing \(t\) by \(f(x)\) gives \[ f(f(x))=a\bigl(f(x)\bigr)+b\bigl(f(x)\bigr)^2 \] Now substitute \(f(x)=ax+bx^2\): \[ f(f(x))=a(ax+bx^2)+b(ax+bx^2)^2 \]

Step 3: Expand the first part.
Let us first simplify \[ a(ax+bx^2)=a^2x+abx^2 \] This part contains only \(x\) and \(x^2\), so it does not directly contribute any \(x^3\) term.

Step 4: Expand the square in the second part.
Now consider \[ (ax+bx^2)^2 \] Using \[ (p+q)^2=p^2+2pq+q^2 \] with \(p=ax\) and \(q=bx^2\), we get \[ (ax+bx^2)^2=a^2x^2+2abx^3+b^2x^4 \]

Step 5: Multiply by \(b\).
Now multiply the above expression by \(b\): \[ b(ax+bx^2)^2=b\left(a^2x^2+2abx^3+b^2x^4\right) \] \[ = a^2bx^2+2ab^2x^3+b^3x^4 \] Here, the coefficient of \(x^3\) is clearly \[ 2ab^2 \]

Step 6: Combine all parts of \(f(f(x))\).
Putting both expanded parts together: \[ f(f(x))=\left(a^2x+abx^2\right)+\left(a^2bx^2+2ab^2x^3+b^3x^4\right) \] \[ f(f(x))=a^2x+\left(ab+a^2b\right)x^2+2ab^2x^3+b^3x^4 \] So the coefficient of \(x^3\) is \[ 2ab^2 \]

Step 7: Final conclusion.
Hence, the required coefficient of \(x^3\) in \(f(f(x))\) is \[ \boxed{2ab^2} \] Therefore, the correct option is \[ \boxed{(5)\ 2ab^2} \]