Question

P is a degree 3 polynomial with p ( 0 ) = 22.08 and zeros 2 and 4 i . Find an equation for p with only real coefficients (i.e. no i in your equation).

Use x as your input, and round all answers to 2 decimal places as needed.

Answer 1

• P(x) = -0.69x³ + 2.76x² - 11.04x + 44.16

Explanation:

Let P(x) is the function with zero's 2 and 4i.

It has to have one more zero since it is of degree 3. This is a conjugate of 4i and is -4i.

The function has the shape of:

• P(x) = a(x - 2)(x - 4i)(x + 4i)

And we have P(0) = 22.08

Let's substitute and find the value of a:

• 22.08 = a(0 - 2)(0 - 4i)(0 + 4i)
• 22.08 = a(-2)(-4i)(4i)
• 22.08 = a(-2)(-16i²)
• 22.08 = a(-32)
• a = - 22.08/32
• a = - 0.69

The function is now:

P(x) = -0.69(x - 2)(x - 4i)(x + 4i)

= -0.69(x - 4)(x² + 16)

= -0.69(x³ - 4x² + 16x - 64)

= -0.69x³ + 2.76x² - 11.04x + 44.16