Write a quartic funciton that has 2 real and 2 imaginary roots and rewrite the polynomialin standard form.
Factor the following quartic equations:
a)x^4-9x^2+18
To write a quartic function with 2 real and 2 imaginary roots, we need to include complex numbers in our equation. The general form of a quartic function is:
f(x) = ax^4 + bx^3 + cx^2 + dx + e
To have 2 real and 2 imaginary roots, we can use the following equation:
f(x) = (x - r)(x - s)(x^2 + bx + c)
Here, r and s are the real roots, and (x^2 + bx + c) represents a quadratic equation with imaginary roots.
To write the polynomial in standard form, we need to expand the above equation:
f(x) = (x - r)(x - s)(x^2 + bx + c)
= (x - r)(x - s)(x^2 + bx + c)
= (x^2 - sx - rx + rs)(x^2 + bx + c)
= x^4 + (b - r - s)x^3 + (c - br - cs)x^2 + (bs + rs^2 - cs^2)x - rcs
So, the quartic function with 2 real and 2 imaginary roots can be written in standard form as:
f(x) = x^4 + (b - r - s)x^3 + (c - br - cs)x^2 + (bs + rs^2 - cs^2)x - rcs