Transform each of the following equations into ones in which of the coefficients of the second highest power of x is zero and also find their transformed equations `x^4+4x^3+2x^2-4x-2=0`

Answer

To transform the equation \( x^4 + 4x^3 + 2x^2 - 4x - 2 = 0 \) into one where the coefficient of the second highest power of \( x \) is zero, we will follow these steps:

### Step 1: Identify the coefficients
The given polynomial can be expressed in the standard form:
\[ a_0 x^4 + a_1 x^3 + a_2 x^2 + a_3 x + a_4 = 0 \]
where:
- \( a_0 = 1 \)
- \( a_1 = 4 \)
- \( a_2 = 2 \)
- \( a_3 = -4 \)
- \( a_4 = -2 \)

### Step 2: Calculate the value of \( h \)
To eliminate the coefficient of \( x^3 \) (which is the second highest power), we use the formula:
\[ h = -\frac{a_1}{n \cdot a_0} \]
where \( n \) is the degree of the polynomial. Here, \( n = 4 \) and \( a_1 = 4 \), \( a_0 = 1 \).

Substituting the values:
\[ h = -\frac{4}{4 \cdot 1} = -1 \]

### Step 3: Perform synthetic division
We will perform synthetic division of the polynomial by \( x + 1 \) (since \( h = -1 \)).

The coefficients are:
\[ 1, \quad 4, \quad 2, \quad -4, \quad -2 \]

Now, we perform synthetic division with \( -1 \):

1. Write down the coefficients:
\[ \begin{array}{r|rrrrr} -1 & 1 & 4 & 2 & -4 & -2 \\ & & -1 & -3 & 1 & 3 \\ \hline & 1 & 3 & -1 & -3 & 1 \\ \end{array} \]

The bottom row gives us the new coefficients:
- \( 1 \) (for \( x^3 \))
- \( 3 \) (for \( x^2 \))
- \( -1 \) (for \( x \))
- \( -3 \) (constant term)

### Step 4: Repeat synthetic division
Now we need to perform synthetic division again with \( -1 \):

1. Write down the new coefficients:
\[ \begin{array}{r|rrrr} -1 & 1 & 3 & -1 & -3 \\ & & -1 & -2 & 3 \\ \hline & 1 & 2 & -3 & 0 \\ \end{array} \]

The new coefficients are:
- \( 1 \) (for \( x^2 \))
- \( 2 \) (for \( x \))
- \( -3 \) (constant term)

### Step 5: Final synthetic division
Perform synthetic division once more with \( -1 \):

1. Write down the new coefficients:
\[ \begin{array}{r|rrr} -1 & 1 & 2 & -3 \\ & & -1 & -1 \\ \hline & 1 & 1 & 0 \\ \end{array} \]

The new coefficients are:
- \( 1 \) (for \( x \))
- \( 1 \) (constant term)

### Step 6: Form the transformed equation
The transformed polynomial is:
\[ x^4 + 0 \cdot x^3 - 4x^2 + 0 \cdot x + 1 = 0 \]
This simplifies to:
\[ x^4 - 4x^2 + 1 = 0 \]

### Final Transformed Equation
The transformed equation is:
\[ x^4 - 4x^2 + 1 = 0 \]