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Understanding the zeros of a polynomial function is crucial because these zeros are the values of the variable that make the function equal to zero. For any polynomial expressed as \( f(x) \), the zeros are the solutions to the equation \( f(x) = 0 \). These zeros can be real or complex numbers. Real zeros are where the graph of the function crosses or touches the x-axis, while complex zeros do not intersect the x-axis and have an imaginary component.
Knowing how to find these zeros helps in understanding the behavior and characteristics of polynomial functions. It is often the first step in solving polynomial equations because they reveal a lot about the function's roots and factors.
To find the zeros of a higher-degree polynomial, we often start by identifying one or more zeros, as this can simplify the process substantially. Each zero corresponds to a factor of the polynomial.

When dealing with complex numbers, you often encounter conjugate pairs. The conjugate of a complex number \( a + bi \) is \( a - bi \), where \( i \) is the imaginary unit \( \sqrt{-1} \). If a polynomial with real coefficients has complex zeros, they will always appear as conjugate pairs. This is because the coefficients are real, and any imaginary parts must cancel out.
In the problem, one zero is \( -3 + \sqrt{2}i \), so its conjugate is \( -3 - \sqrt{2}i \). These pairs help simplify the polynomial division process in solving for other zeros. They ensure that when multiplying conjugates, the product remains a real number, eliminating the imaginary component and simplifying calculations.
This is particularly useful because it means that we can set up factors of the polynomial function corresponding to these conjugate pairs without retaining complex components.

The quadratic formula is a powerful tool in solving quadratic equations. Quadratics are polynomials of degree two, usually expressed in the form \( ax^2 + bx + c = 0 \). The quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) allows us to find the zeros of the quadratic equation, where \( a \), \( b \), and \( c \) are coefficients.
Using this formula is straightforward. Start by identifying the coefficients \( a \), \( b \), and \( c \) from the quadratic expression. Then, substitute these into the formula: calculate the discriminant \( b^2 - 4ac \) and proceed to find the potential two solutions (as indicated by the \( \pm \) symbol), which may be real or complex.
In the given problem, after using polynomial division, we derived a quadratic expression \( x^2 - 3x + 2 \). Plugging into the quadratic formula gives us the zeros \( x = 1 \) and \( x = 2 \), thereby completing the process of finding all zeros.

Polynomial division is a method similar to traditional long division, and it helps simplify polynomial expressions, particularly when seeking zeros of higher-degree polynomials. It's an essential step when a polynomial expression can be factored further given known zeros.
In the context of the exercise, we first use the identified complex conjugate zeros to form a quadratic factor, \( x^2 + 6x + 11 \). We then divide the original polynomial \( f(x) \) by this factor. This process reduces the polynomial's degree, allowing simpler methods like the quadratic formula to be used on the quotient.
During division, you repeatedly subtract the product of the divisor and a term from the dividend until no further division is possible. This reveals a new polynomial of lower degree, which was \( x^2 - 3x + 2 \) in this case, simplifying further analysis to find the remaining zeros.