Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=x^4-6x^3+7x^2

Verified step by step guidance

1. <insert step 1> Start by factoring out the greatest common factor from the polynomial \( f(x) = x^4 - 6x^3 + 7x^2 \). Notice that \( x^2 \) is a common factor in all terms, so factor it out: \( f(x) = x^2(x^2 - 6x + 7) \).
2. <insert step 2> Now, solve for the zeros of the factored polynomial. First, set the factored expression equal to zero: \( x^2(x^2 - 6x + 7) = 0 \).
3. <insert step 3> Solve \( x^2 = 0 \) to find the first set of zeros. This gives \( x = 0 \). Since \( x^2 \) is a factor, \( x = 0 \) is a zero with multiplicity 2.
4. <insert step 4> Next, solve the quadratic equation \( x^2 - 6x + 7 = 0 \) using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -6 \), and \( c = 7 \).
5. <insert step 5> Calculate the discriminant \( b^2 - 4ac \) to determine the nature of the roots. If the discriminant is negative, the roots will be complex. Substitute the values into the quadratic formula to find the complex zeros.

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 general form of a polynomial in one variable is f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where n is a non-negative integer and a_n is not zero. Understanding the structure of polynomial functions is essential for analyzing their roots and behavior.

Complex Zeros

Complex zeros of a polynomial are the values of x for which the polynomial evaluates to zero, and they can be real or 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 techniques such as factoring, synthetic division, or applying the quadratic formula.

Factoring and the Rational Root Theorem

Factoring is the process of breaking down a polynomial into simpler components (factors) that can be multiplied to yield the original polynomial. The Rational Root Theorem provides a method to identify possible rational roots of a polynomial, which can then be tested to find actual roots. This theorem is particularly useful for polynomials with integer coefficients, as it helps narrow down the search for zeros.