Question

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=x4-8x3+29x2-66x+72

Accepted Answer

Start by examining the polynomial function $$f(x) = x^4$ - 8x^3$ + 29x^2$ - 66x + 72 to $$find its complex zeros. Since it is a quartic (degree 4), there will be 4 zeros in total, counting multiplicities and complex roots. Attempt to find rational roots using the Rational Root Theorem. The possible rational roots are factors of the constant term 72 divided by factors of the leading coefficient 1, so test $$\pm1, \pm2, \pm3, \pm4, \pm6, \pm8, \pm9, \pm12, \pm18, \pm24, \pm36, \pm72 by $$substituting into $$f(x). $$Once a root $$r is $$found, use polynomial division (either synthetic or long division) to divide $$f(x) by$$ $$(x - r)$$, reducing the quartic to a cubic polynomial. Repeat the process of finding roots and factoring the polynomial until it is factored completely into linear and/or quadratic factors. If a quadratic factor remains that cannot be factored further over the reals, use the quadratic formula to find its complex zeros. Write down all zeros found from the linear factors and the solutions from the quadratic formula, ensuring to express the complex zeros in exact form (using $$i$$ for the imaginary unit).

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=x⁴-8x³+29x²-66x+72

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Complex Zeros of Polynomial Functions

Polynomial Factoring and Synthetic Division

Quadratic Formula for Solving Quadratic Equations