In mathematics, a factor of a polynomial is an expression that divides the polynomial evenly, leaving no remainder. When a polynomial is set equal to zero, the solutions to that equation are called the zeros or roots of the polynomial. The relationship between factors and zeros is fundamental:
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- If (x - a) is a factor of a polynomial, then x = a is a zero of the polynomial.
- Conversely, if x = a is a zero of a polynomial, then (x - a) is a factor of the polynomial.
This concept is often illustrated using the Factor Theorem:
Factor Theorem: For a polynomial f(x), (x - a) is a factor of f(x) if and only if f(a) = 0.
Complex Conjugate Roots
In the case of polynomials with real coefficients, complex roots always occur in conjugate pairs. This means if (x + bi) is a root, then its conjugate (x - bi) is also a root.
Analyzing the Given Factors
The given function, f(x), has the following factors:
- (x - 7): This factor indicates that x = 7 is a real zero of f(x).
- (x + 4i): This factor indicates that x = -4i is a complex zero of f(x). Since the coefficients of f(x) are real, the complex conjugate, x = 4i, must also be a zero of f(x).
Finding the Total Number of Real Zeros
We have identified one real zero (x = 7) and two complex zeros (x = -4i and x = 4i). Since complex zeros always occur in conjugate pairs, the total number of real zeros of f(x) is **1**.
It's important to note that we cannot definitively determine the degree of the polynomial f(x) with only these factors. It's possible that f(x) has other real or complex factors.
Example:
Let's say f(x) is a third-degree polynomial. Since it has (x-7) and (x+4i) as factors, it must also have (x-4i) as a factor to satisfy the conjugate pair property. The polynomial would then be:
f(x) = (x-7)(x+4i)(x-4i)
Expanding this, we get:
f(x) = (x-7)(x^2 + 16)
This polynomial has one real zero (x = 7) and two complex zeros (x = 4i and x = -4i).
Key Takeaways
This example illustrates the fundamental concepts of factors and zeros in polynomial functions:
- Factors determine the zeros of a polynomial.
- Real polynomials with complex zeros have these zeros in conjugate pairs.
- Knowing some factors of a polynomial allows us to deduce some of its zeros, but it doesn't always provide information about the polynomial's degree or all its zeros.