what does it mean when two of the zeros of a polynomial function are complex conjugates?
x^2=y+2
When two of the zeros of a polynomial function are complex conjugates, it means that these two zeros have the form a + bi and a - bi, where a and b are real numbers, and i is the imaginary unit (√-1). Complex conjugates always come in pairs.
To understand this concept further, we need to recall the fundamental theorem of algebra, which states that every polynomial equation has exactly as many complex roots (zeros) as its degree.
Let's assume we have a polynomial function in the form f(x) = a(x - r)(x - s)(x - t), where r and s are the complex conjugate zeros and t is another real or complex zero. This polynomial can be written as f(x) = a(x - r)(x - s)(x - t) = a(x^2 - (r + s)x + rs)(x - t).
Now, by expanding and collecting similar terms, we get f(x) = a(x^3 - (r + s + t)x^2 + (rs + rt + st)x - rst).
From the equation, it is clear that the coefficients of the polynomial are real, and thus, for every complex zero r + si, its conjugate r - si must also be a zero. This property is due to the fact that the coefficients of the polynomial are real numbers, and when we multiply a complex conjugate with its conjugate, the imaginary parts cancel out, resulting in a real number.
In summary, when two zeros of a polynomial function are complex conjugates, it indicates that the polynomial has real coefficients, and the complex conjugate zeros occur in pairs to maintain the real coefficients.