Question

Write a polynomial in factored form that has roots r = -4, 7, and 1 = i√2.

Answer 1

The polynomial in factored form with roots
`\(r = -4, 7, \text{ and } 1 + i√(2)\)` is given by:

`\[ P(x) = (x + 4)(x - 7)(x - 1 - i√(2))(x - 1 + i√(2)) \]`

Step-by-step explanation:

To find the polynomial in factored form with the given roots, we use the fact that if \(r\) is a root of a polynomial, then \((x - r)\) is a factor of that polynomial. In this case, the roots are
`\(r = -4, 7, \text{ and } 1 + i√(2)\).`Therefore, the corresponding factors are
`\((x + 4), (x - 7), (x - 1 - i√(2))\), and \((x - 1 + i√(2))\)`. Multiplying these factors together gives us the desired polynomial.

Now, let's expand and simplify the expression:

`\[ P(x) = (x + 4)(x - 7)(x - 1 - i√(2))(x - 1 + i√(2)) \]`

Expanding the first two factors:

`\[ P(x) = (x^2 - 3x - 28)(x - 1 - i√(2))(x - 1 + i√(2)) \]`

Multiplying the remaining factors:

`\[ P(x) = (x^2 - 3x - 28)((x - 1)^2 - (i√(2))^2) \]`

Simplifying further:

`\[ P(x) = (x^2 - 3x - 28)(x^2 - 2x + 1 + 2) \]`

Combining like terms:

`\[ P(x) = (x^2 - 3x - 28)(x^2 - 2x + 3) \]`

This is the polynomial in factored form with the given roots.