Question

Find a degree 3 polynomial with real coefficients having zeros 3 and 4 i and a lead coefficient of 1. Write P in expanded form. Be sure to write the full equation, including P ( x ) = .

Answer 1

`P(x)=x^3-3x^2+16x-48`

Explanation:

Given information:

• Polynomial function with real coefficients.
• Zeros: 3 and 4i
• Lead coefficient of 1.

For any complex number
`z=a+bi`, the complex conjugate of the number is defined as
`z^*=a-bi`.

If f(z) is a polynomial with real coefficients, and z₁ is a root of f(z)=0, then its complex conjugate z₁* is also a root of f(z)=0.

Therefore, if P(x) is a polynomial with real coefficients, and 4i is a root of f(x)=0, then its complex conjugate -4i is also a root of P(x)=0.

Therefore, the polynomial in factored form is:

`P(x)=1(x-3)(x-4i)(x-(-4i))`

`P(x)=(x-3)(x-4i)(x+4i)`

Expand the polynomial:

`P(x)=(x-3)(x^2+4ix-4ix-16i^2)`

`P(x)=(x-3)(x^2-16(-1))`

`P(x)=(x-3)(x^2+16)`

`P(x)=x^3+16x-3x^2-48`

`P(x)=x^3-3x^2+16x-48`