Question

When each root of the equation \[ x^3-ax^2+bx+1=0 \] is diminished by \(h\), if the transformed equation takes the form \[ 9x^3+\left(9+\frac{a^3}{3}\right)=0, \] then \(ah=\)

• A. \(b\)
• B. \(1\)
• C. \(a^2+b\)
• D. \(\dfrac{b^2}{3}\)

Hint:

If every root of a polynomial is decreased by \(h\), replace \[ \boxed{x\rightarrow x+h.} \] After expansion, compare coefficients to determine the required value.

Answer 1

The Correct Option is A

Step 1: Form the transformed equation. If each root is diminished by \(h\), then replace \[ x\rightarrow x+h. \] Hence, \[ (x+h)^3-a(x+h)^2+b(x+h)+1=0. \] Expanding, \[ x^3+(3h-a)x^2+\left(3h^2-2ah+b\right)x +\left(h^3-ah^2+bh+1\right)=0. \]

Step 2: Compare with the given transformed equation. The transformed equation is \[ 9x^3+\left(9+\frac{a^3}{3}\right)=0. \] Dividing throughout by \(9\), \[ x^3+\frac{9+\frac{a^3}{3}}9=0. \] Since there are no \(x^2\) and \(x\) terms, \[ 3h-a=0, \] \[ 3h^2-2ah+b=0. \] From the first equation, \[ \boxed{h=\frac{a}{3}.} \] Substituting into the second equation, \[ 3\left(\frac{a}{3}\right)^2 - 2a\left(\frac{a}{3}\right) +b=0, \] \[ \frac{a^2}{3} - \frac{2a^2}{3} +b=0, \] \[ \boxed{b=\frac{a^2}{3}.} \]

Step 3: Find \(ah\). Using \[ h=\frac{a}{3}, \] we get \[ ah = a\left(\frac{a}{3}\right) = \frac{a^2}{3} = b. \] Hence, \[ \boxed{ah=b.} \] Thus, \[ \boxed{(A)} \] is the correct answer.