Remove second term ( second higher power of x ) from the equation
` x^4 +8x^3 +x-5=0`

Answer

To remove the second term (the second highest power of \(x\)) from the equation \(x^4 + 8x^3 + x - 5 = 0\), we can use a substitution method. Here’s the step-by-step solution:

### Step 1: Substitute \(x\) with \(y + a\)
We will make the substitution \(x = y + a\), where \(a\) is a constant that we will determine later. This substitution will help us eliminate the \(x^3\) term.

### Step 2: Expand the equation
Substituting \(x = y + a\) into the equation gives us:
\[ (y + a)^4 + 8(y + a)^3 + (y + a) - 5 = 0 \]

### Step 3: Expand each term
Now we expand each term:
- \((y + a)^4 = y^4 + 4ay^3 + 6a^2y^2 + 4a^3y + a^4\)
- \((y + a)^3 = y^3 + 3ay^2 + 3a^2y + a^3\)

Thus, we can rewrite the equation as:
\[ y^4 + 4ay^3 + 6a^2y^2 + 4a^3y + a^4 + 8(y^3 + 3ay^2 + 3a^2y + a^3) + (y + a) - 5 = 0 \]

### Step 4: Combine like terms
Combining all the terms, we have:
\[ y^4 + (4a + 8)y^3 + (6a^2 + 24a)y^2 + (4a^3 + 24a^2 + 1)y + (a^4 + 8a^3 + a - 5) = 0 \]

### Step 5: Set the coefficient of \(y^3\) to zero
To remove the \(y^3\) term, we set the coefficient of \(y^3\) to zero:
\[ 4a + 8 = 0 \]
Solving for \(a\):
\[ 4a = -8 \implies a = -2 \]

### Step 6: Substitute \(a\) back into the equation
Now substitute \(a = -2\) back into the equation:
\[ y^4 + (6(-2)^2 + 24(-2))y^2 + (4(-2)^3 + 24(-2)^2 + 1)y + ((-2)^4 + 8(-2)^3 - 2 - 5) = 0 \]

### Step 7: Simplify the coefficients
Calculating the coefficients:
- For \(y^2\):
\[ 6(4) + 24(-2) = 24 - 48 = -24 \]
- For \(y\):
\[ 4(-8) + 24(4) + 1 = -32 + 96 + 1 = 65 \]
- For the constant term:
\[ 16 - 64 - 2 - 5 = 16 - 64 - 7 = -55 \]

### Final Equation
Thus, the transformed equation is:
\[ y^4 - 24y^2 + 65y - 55 = 0 \]

### Summary
We have successfully removed the second highest power of \(x\) from the original equation.

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