Question

REASONING Two zeros of \(f(x)=x^3-6 x^2-16 x+96\) are 4 and \(-4\). Explain why the third zero must also be a real number.

Short answer

The third root must be real because cubic polynomials have three roots and complex roots, if present, appear in conjugate pairs. As the given roots, 4 and -4, are real, it's impossible to have a complex conjugate pair, so the remaining root must also be real.

Step-by-step solution

Understanding the fundamental theorem of algebra

The fundamental theorem of algebra states that every nonzero, single-variable, degree-n polynomial with complex coefficients has, counted with multiplicity, exactly n complex roots. This theorem implies that a polynomial f(x) of degree n will always have n roots or zeros.

Knowing complex roots come in conjugate pairs

Complex roots of polynomials come in conjugate pairs when the coefficients are real numbers. Thus, if a+bi is a root, its conjugate, a-bi, must also be a root.

Putting it all together

Given that this is a cubic polynomial, we know that it should have three zeros. We are already given two roots, 4 and -4. Since both of these roots are real numbers, we are left with one more root. It can't be a complex number because complex roots must appear in conjugate pairs. Therefore, the third root also must be a real number.

Key concepts

Fundamental Theorem of Algebra

Understanding the fundamental theorem of algebra is crucial for solving polynomial equations. This pivotal theorem asserts that every non-zero polynomial equation with complex coefficients and a degree of n has exactly n roots, if each root is counted with its multiplicity. Multiplicity refers to the number of times a particular root occurs. For instance, if a polynomial has a root of 2 that appears twice, it has a multiplicity of two.

In simpler terms, this means that a quadratic polynomial (degree 2) is guaranteed two roots, a cubic polynomial (degree 3) will have three roots, and so on. These roots can be real or complex numbers. A key takeaway is that if you're working with a polynomial with real coefficients, as most textbook problems present, and you've found some, but not all, of the roots, the fundamental theorem guides you on how many more roots you need to find.

Complex Conjugate Roots

Polynomials with real coefficients showcase an interesting behavior when it comes to their complex roots—they always come in pairs, known as complex conjugate roots. For a given complex root in the form of \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit \(\sqrt{-1}\), there will always be another root that is its conjugate, \(a - bi\).

Why is this important? When you're given a polynomial equation and asked to find its roots, if you discover a complex root, you can immediately know there's another root which is its conjugate. This pairs-up concept helps you identify all the roots of a polynomial with real coefficients. If a polynomial of degree n has a single complex root, it cannot satisfy the condition of having n roots without its complex conjugate pair being included.

Cubic Polynomials

Cubic polynomials are third-degree polynomials, characterized by their highest exponent being three. An example is \(f(x)=ax^3+bx^2+cx+d\), where \(a\), \(b\), \(c\), and \(d\) are coefficients and \(a \eq 0\). The shape of the graph of a cubic polynomial can vary widely, but it will always have at least one real root due to the Intermediate Value Theorem — which essentially states that a continuous function that takes on different signs at two points must cross the horizontal axis at least once between those points.

Given that a cubic polynomial must have exactly three roots based on the fundamental theorem of algebra, finding all the zeros can be a matter of not just computation but also understanding these underlying principles. If you're provided with two real zeros, as in our exercise, then the third must also be real due to the inability to form a complex conjugate pair, which keeps the total count of roots to three, fulfilling the fundamental theorem’s criteria.