find the (monic ) polynomial equation of lowest degree whose roots are
`1,3,5,7`

Answer

To find the monic polynomial equation of the lowest degree whose roots are 1, 3, 5, and 7, we can follow these steps:

### Step 1: Write the roots in terms of factors
Since the roots of the polynomial are given as 1, 3, 5, and 7, we can express the polynomial as:
\[ P(x) = (x - 1)(x - 3)(x - 5)(x - 7) \]

### Step 2: Multiply the factors in pairs
We can multiply the factors in pairs to simplify the calculation. First, we can multiply \((x - 1)\) and \((x - 7)\):
\[ (x - 1)(x - 7) = x^2 - 8x + 7 \]

Next, we multiply \((x - 3)\) and \((x - 5)\):
\[ (x - 3)(x - 5) = x^2 - 8x + 15 \]

### Step 3: Multiply the resulting quadratic polynomials
Now we have two quadratic polynomials:
1. \(x^2 - 8x + 7\)
2. \(x^2 - 8x + 15\)

We will multiply these two quadratics:
\[ P(x) = (x^2 - 8x + 7)(x^2 - 8x + 15) \]

### Step 4: Use substitution to simplify multiplication
Let \(y = x^2 - 8x\). Then, we can rewrite the polynomial as:
\[ P(x) = (y + 7)(y + 15) \]

### Step 5: Expand the product
Now we can expand:
\[ P(x) = y^2 + 15y + 7y + 105 = y^2 + 22y + 105 \]

### Step 6: Substitute back for \(y\)
Now substitute back \(y = x^2 - 8x\):
\[ P(x) = (x^2 - 8x)^2 + 22(x^2 - 8x) + 105 \]

### Step 7: Expand the expression
Now we expand \((x^2 - 8x)^2\):
\[ (x^2 - 8x)^2 = x^4 - 16x^3 + 64x^2 \]
Now combine all parts:
\[ P(x) = x^4 - 16x^3 + 64x^2 + 22x^2 - 176x + 105 \]
Combine like terms:
\[ P(x) = x^4 - 16x^3 + (64 + 22)x^2 - 176x + 105 \]
\[ P(x) = x^4 - 16x^3 + 86x^2 - 176x + 105 \]

### Final Polynomial
Thus, the monic polynomial equation of the lowest degree whose roots are 1, 3, 5, and 7 is:
\[ P(x) = x^4 - 16x^3 + 86x^2 - 176x + 105 = 0 \]