Find all the zeros of each equation x^5-3x^4-15x^3+45x^2-16x+48=0

The zeros of the equation [tex]{x^5}-3{x^4}-15{x^3}+45{x^2}-16x+48=0[/tex] are [tex]\boxed{--4,{\text{ }}3,{\text{ }}4,+i{\text{ and}}-i}[/tex] Further explanation: The Fundamental Theorem of Algebra states that the polynomial has n roots if the degree of the polynomial is n. [tex]f\left(x\right)=a{x^n}+b{x^{n-1}}+\ldots+cx+d[/tex] The polynomial function has n roots or zeroes. Given: The polynomial function is [tex]{x^5}-3{x^4}-15{x^3}+45{x^2}-16x+48=0[/tex]. Explanation: The polynomial function [tex]{x^5}-3{x^4}-15{x^3}+45{x^2}-16x+48=0[/tex] has five zeroes as the degree of the polynomial is 5. Solve the equation to obtain the zeroes. [tex]\begin{aligned}\left({{x^5}-3{x^4}}\right)+\left({-15{x^3}+45{x^2}}\right)+\left({-16x+48}\right)&=0\\{x^4}\left({x-3}\right)-15{x^2}\left({x-3}\right)-16\left({x-3}\right)&=0\\\left({x-3}\right)\left({{x^4}-15{x^2}-16}\right)&=0\\\end{aligned}[/tex] The first zero of the equation can be calculated as follows, [tex]\begin{aligned}x-3&=0\\x&=3\\\end{aligned}[/tex] Now solve the equation [tex]{x^4}-15{x^2}[/tex]-16 to obtain the remaining zeros. [tex]\begin{aligned}{x^4}-15{x^2}-16&=0\\\left({{x^2}-16}\right)\left({{x^2}+1}\right)&=0\\\end{aligned}[/tex] Substitute [tex]{x^2}[/tex]-16 equal to zero to obtain the zeroes. [tex]\begin{aligned}{x^2}-16&=0\\{x^2}&=16\\x&=\sqrt{16}\\x&=\pm4\\\end{aligned}[/tex] There zeroes are 4 and -4. Now substitute [tex]{x^2}[/tex]+1 equal to zero to obtain the zeros. [tex]\begin{aligned}{x^2}+1&=0\\{x^2}&=-1\\x&=\sqrt{-1}\\x&=\pm i\\\end{aligned}[/tex] Hence, the zeros of the equation [tex]{x^5}-3{x^4}-15{x^3}+45{x^2}-16x+48=0[/tex] are [tex]\boxed{--4,{\text{ }}3,{\text{ }}4,+i{\text{ and}}-i}[/tex] Learn more: 1. Learn more about inverse of the function . 2. Learn more about equation of circle brainly.com/question/1506955. 3. Learn more about range and domain of the function Answer details: Grade: High School Subject: Mathematics Chapter: Polynomials Keywords: quadratic equation, equation factorization. Factorized form, polynomial, quadratic formula, zeroes, Fundamental Theorem of algebra, polynomial, six roots.