Question

Find the multiplicity of the zeros of the polynomial of f(x)=x^4-2x^3+x^2

Answer 1

0 multiplicity 2 and 1 multiplicity 2.

Explanation:

`f(x)=x^4-2x^3+x^2\\\\f(x)=x^2(x^2-2x+1)=x^2(x-1)^2\\\\/\text{used}\ (a-b)^2=a^2-2ab+b^2/\\\\\text{The zeros:}\\\\f(x)=0\iff x^2(x-1)^2=0\\\\x^2=0\ \vee\ (x-1)^2=0\\\\x=0\ \vee\ x-1=0\\\\x=0\ \vee\ x=1`

`\text{The zero}\ x =0\ \text{has multiplicity 2 because the factor}\\x\ \text{occurs two times (that is, the factor is raised to the second power)}.\\\\\text{The zero}\ x =1\ \text{has multiplicity 2 because the factor}\\(x-1)\ \text{occurs two times (that is, the factor is raised to the second power)}.`