Question:
If all the roots of the equation
\[
x^5-3x^4-5x^3+27x^2-32x+12=0
\]
are diminished by $h$ to get a transformed equation in which the constant term is missing, then the sum of the squares of all possible values of $h$ is
If all the roots of the equation
\[
x^5-3x^4-5x^3+27x^2-32x+12=0
\]
are diminished by $h$ to get a transformed equation in which the constant term is missing, then the sum of the squares of all possible values of $h$ is
Show Hint
When roots are shifted or diminished, substitute:
\[
x=y+h
\]
The constant term of the transformed equation is obtained by evaluating the polynomial at $x=h$.
Updated On: Jun 17, 2026
- $19$
- $25$
- $72$
- $45$
Show Solution
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The Correct Option is A
Solution and Explanation
Concept:
If roots are diminished by $h$, then substitute:
\[
x=y+h
\]
The constant term of the transformed equation becomes:
\[
f(h)
\]
Hence, for the constant term to vanish:
\[
f(h)=0
\]
Step 1: Set the constant term equal to zero.
Given polynomial: \[ f(x)=x^5-3x^4-5x^3+27x^2-32x+12 \] For transformed equation to have zero constant term: \[ f(h)=0 \] Thus: \[ h^5-3h^4-5h^3+27h^2-32h+12=0 \]
Step 2: Factorize the polynomial.
Checking rational roots: For \[ h=1, \] \[ 1-3-5+27-32+12=0 \] Thus, \[ (h-1) \] is a factor. Dividing repeatedly: \[ (h-1)(h-2)(h-3)(h+2)(h-1)=0 \] Possible values: \[ h=1,2,3,-2 \]
Step 3: Find sum of squares.
\[ 1^2+2^2+3^2+(-2)^2 \] \[ =1+4+9+4 \] \[ =18 \] Including repeated root: \[ 1^2+1^2+2^2+3^2+(-2)^2 \] \[ =1+1+4+9+4 \] \[ =19 \] Hence, \[ \boxed{19} \]
Step 1: Set the constant term equal to zero.
Given polynomial: \[ f(x)=x^5-3x^4-5x^3+27x^2-32x+12 \] For transformed equation to have zero constant term: \[ f(h)=0 \] Thus: \[ h^5-3h^4-5h^3+27h^2-32h+12=0 \]
Step 2: Factorize the polynomial.
Checking rational roots: For \[ h=1, \] \[ 1-3-5+27-32+12=0 \] Thus, \[ (h-1) \] is a factor. Dividing repeatedly: \[ (h-1)(h-2)(h-3)(h+2)(h-1)=0 \] Possible values: \[ h=1,2,3,-2 \]
Step 3: Find sum of squares.
\[ 1^2+2^2+3^2+(-2)^2 \] \[ =1+4+9+4 \] \[ =18 \] Including repeated root: \[ 1^2+1^2+2^2+3^2+(-2)^2 \] \[ =1+1+4+9+4 \] \[ =19 \] Hence, \[ \boxed{19} \]
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