Solve each problem. A comprehensive graph of ƒ(x)=x^4-7x^3+18x^2-22x+12 is shown in the two screens, along with displays of the two real zeros. Find the two remaining nonreal complex zeros.

Verified step by step guidance

1. Identify the given polynomial function: \( f(x) = x^4 - 7x^3 + 18x^2 - 22x + 12 \).
2. Recognize that the polynomial is of degree 4, which means it has 4 roots (real or complex).
3. Note that the problem states there are two real zeros, so the remaining two zeros must be nonreal complex numbers.
4. Use the fact that complex roots of polynomials with real coefficients come in conjugate pairs. Therefore, if \( a + bi \) is a root, \( a - bi \) is also a root.
5. Apply the Fundamental Theorem of Algebra and the Conjugate Root Theorem to find the two nonreal complex zeros, considering the given real zeros and the degree of the polynomial.

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In this case, ƒ(x) = x^4 - 7x^3 + 18x^2 - 22x + 12 is a fourth-degree polynomial, which can have up to four roots (zeros). Understanding the behavior of polynomial functions, including their degree and leading coefficient, is essential for analyzing their graphs and finding their zeros.

Complex Zeros

Complex zeros are solutions to polynomial equations that are not real numbers. They occur in conjugate pairs due to the Fundamental Theorem of Algebra, which states that a polynomial of degree n has exactly n roots in the complex number system. Identifying complex zeros involves recognizing that if a polynomial has real coefficients, any nonreal zeros must appear as pairs of the form a + bi and a - bi, where a and b are real numbers.

Graphical Interpretation of Zeros

The zeros of a polynomial function correspond to the x-intercepts of its graph. By analyzing the graph of ƒ(x), one can visually identify the real zeros and infer the presence of complex zeros when the graph does not intersect the x-axis at certain points. Understanding how to interpret the graph helps in determining the nature and number of zeros, including distinguishing between real and complex solutions.