Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=x⁴-6x³+7x²

Verified step by step guidance

1. Start by writing down the polynomial function: \(f(x) = x^4 - 6x^3 + 7x^2\).
2. Factor out the greatest common factor (GCF) from all terms. Since each term contains \(x^2\), factor it out: \(f(x) = x^2(x^2 - 6x + 7)\).
3. Set the factored form equal to zero to find the zeros: \(x^2(x^2 - 6x + 7) = 0\). This implies either \(x^2 = 0\) or \(x^2 - 6x + 7 = 0\).
4. Solve the first equation \(x^2 = 0\) to find the zeros from this factor. Then, 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. Simplify the solutions from the quadratic formula to find the exact complex zeros. Combine these with the zeros from \(x^2=0\) to list all complex zeros of the polynomial.

Key Concepts

Polynomial Zeros

Polynomial zeros are the values of x for which the polynomial equals zero. Finding zeros involves solving the equation f(x) = 0. These zeros can be real or complex numbers and represent the roots of the polynomial function.

Factoring Polynomials

Factoring is the process of expressing a polynomial as a product of simpler polynomials. It helps in finding zeros by setting each factor equal to zero. Techniques include factoring out common terms, grouping, or using special formulas like difference of squares.

Complex Numbers and the Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that a polynomial of degree n has exactly n roots in the complex number system, counting multiplicities. Complex numbers include real and imaginary parts, allowing solutions beyond real zeros when factoring over the reals is not possible.