Given a polynomial that has zeros of - 4, 9i, and - 9i and has a value of 492 when x=-1. Write the polynomial in standard form axn+bxn-1+ ... Answer using reduced fractions when necessary.

The polynomial that has zeros of - 4, 9i, and - 9i, and has a value 492 when x = - 1 is:

2x³ + 8x² + 162x + 648 = 0

Step-by-step explanation:

Given that the polynomial that has zeros of - 4, 9i, and - 9i, we can say that

x = - 4 = > x + 4 = 0

x = 9i and - 9i = > x = ±9i = > x² = (9i) ²

Because i² = - 1, x² = - 81

=> x² + 81 = 0

So together, we can write the polynomial as

(x + 4) (x² + 81) = 0

Expanding the bracket, we have

x³ + 81x + 4x² + 324 = 0

x³ + 4x² + 81x + 324 = 0

Again, we are told that the polynomial is 492 at x = - 1.

Put x = - 1 in

x³ + 4x² + 81x + 324 = 0

(-1) ³ + 4 (-1) ² + 81 (-1) + 324

-1 + 4 - 81 + 324

= 246 ≠ 492

But 492 = 2*246

Since multiplying the polynomial by 2 doesn't change anything, our result is set.

Now,

2 (x³ + 4x² + 81x + 324) = 2 (0)

2x³ + 8x² + 162x + 648 = 0

When x = - 1

2 (-1) ³ + 8 (-1) ² + 162 (-1) + 648

-2 + 8 - 162 + 648

= 492

As required.