Question

Find the zeros of the polynomial function and state the multiplicity of each.

Answer 1

$\bf f(x) = 4(x + 7)^2(x - 7)^3\implies 0 = 4(x + 7)^2(x - 7)^3 \\\\\\ 0=(x+7)^2(x-7)^3\implies \begin{cases} 0=(x+7)^7\\ \qquad 0=(x+7)(x+7)\\ \qquad -7=x\\ \qquad -7=x\\ ----------\\ 0=(x-7)^3\\ \qquad 0=(x-7)(x-7)(x-7)\\ \qquad 7=x\\ \qquad 7=x\\ \qquad 7=x \end{cases}$

the -7 is found there twice, so it has a multiplicity of 2
the +7 is there thrice, so it has multiplicity of 3