Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=x^6-9x^4-16x^2+144

Verified step by step guidance

1. <Step 1: Recognize the polynomial structure.> The given polynomial is $f(x) = x^6 - 9x^4 - 16x^2 + 144$. Notice that the powers of $x$ are all even, which suggests a substitution might simplify the problem.
2. <Step 2: Make a substitution.> Let $u = x^2$. Then, the polynomial becomes $u^3 - 9u^2 - 16u + 144$. This is a cubic polynomial in terms of $u$.
3. <Step 3: Solve the cubic equation.> Use methods such as factoring, synthetic division, or the Rational Root Theorem to find the roots of the cubic equation $u^3 - 9u^2 - 16u + 144 = 0$. These roots will be the values of $u = x^2$. Let's denote these roots as $u_1, u_2, u_3$.
4. <Step 4: Find $x$ from $u$.> For each root $u_i$, solve the equation $x^2 = u_i$ to find the corresponding values of $x$. This will involve taking the square root of each $u_i$, giving $x = \pm\sqrt{u_i}$.
5. <Step 5: List all complex zeros.> Combine all the solutions from Step 4 to list all complex zeros of the original polynomial $f(x)$. Remember to include both positive and negative square roots for each $u_i$.

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The degree of the polynomial is determined by the highest power of the variable. Understanding the structure of polynomial functions is essential for analyzing their roots, which can be real or complex.

Complex Zeros

Complex zeros of a polynomial are the values of the variable that make the polynomial equal to zero, which can include both real and non-real (complex) numbers. According to the Fundamental Theorem of Algebra, a polynomial of degree n has exactly n roots in the complex number system, counting multiplicities. Identifying complex zeros often involves factoring or using the quadratic formula.

Factoring Polynomials

Factoring polynomials involves expressing the polynomial as a product of simpler polynomials or linear factors. This process is crucial for finding the zeros of the polynomial, as the zeros correspond to the values that make each factor equal to zero. Techniques such as synthetic division, grouping, or applying the Rational Root Theorem can aid in the factoring process.