Question

We say that $\mathit{a}\mathbf{\in }\mathit{F}$is a multiple root of $\mathit{f}\left(x\right)\mathbf{\in }\mathit{F}\left[x\right]$is a factor of f (x) for some $\mathit{k}\mathbf{⩾}\mathbf{2}$.

(a) Prove that $\mathit{a}\mathbf{\in }\mathbf{ℝ}$is a multiple root of $\mathit{f}\left(x\right)\mathbf{\in }\mathbf{ℝ}\left[x\right]$if and only if a is a root of both f (x) and f'(x), where f'(x) is the derivative of f (x).

(b) If $\mathit{f}\left(x\right)\mathbf{\in }\mathbf{ℝ}\left[x\right]$and if f (x) is relatively prime to f'(x). Prove that has no multiple f(x) root in role="math" localid="1648662770183" $\mathbf{ℝ}$.

Short answer

It is proved that

1. a is a multiple root of f (x).
2. f (x) has no multiple roots.

Step-by-step solution

a)Step 1: Determine a∈ℝ is a multiple root of f(x)∈ℝ[x]

Consider the given functions, $a\in \mathrm{ℝ}$is a multiple root of $f\left(x\right)\in F\left[x\right]$. Thus, it can be written as,

$f\left(x\right)={\left(x-a\right)}^{2}h\left(x\right)$for $h\left(x\right)\in \mathrm{ℝ}\left[x\right]$. Along with the definition, a is a root of f (x). It contain that,

$f\text{'}\left(x\right)=2\left(x-a\right)h\left(x\right)+{\left(x-a\right)}^{2}h\text{'}\left(x\right)=\left(x-a\right)\left(2h\left(x\right)+\left(x-a\right)h\text{'}\left(x\right)\right)$

From above equation, we can be conclude that,

$\left.\left(x-a\right)\right|f\text{'}\left(x\right)$. Therefore, a is root of f'(x).

Now, assume that is a root of both f(x) and f'(x). Therefore, $h\left(x\right)\in \mathrm{ℝ}\left[x\right]$then, $f\left(x\right)=\left(x-a\right)h\left(x\right)$. Similarly, $f\text{'}\left(x\right)=h\left(x\right)+\left(x-a\right)h\text{'}\left(x\right)$.

Consider that, $f\text{'}\left(a\right)=0$, thus, $h\left(a\right)=0$and hence, there is some $k\left(x\right)\in \mathrm{ℝ}\left[x\right]$so that, $h\left(x\right)=\left(x-a\right)k\left(x\right)$, and $f\left(x\right)={\left(x-a\right)}^{2}k\left(x\right)$, thus is a multiple root of f (x).

b)Step 2: Determine no multiple f (x) root in ℝ

If f (x) and f'(x) are relatively prime then, these share no linear factors and roots.

It means that f (x) has no multiple roots.