Properties of Polynomials Examples 1

Recall from the Properties of Polynomials page that a function of the form $p(x) = a_0 + a_1x + a_2x^2 + ... + a_nx^n$ is called a polynomial, and if $a_m \neq 0$ then we say that the degree of the polynomial $p$ is $n$ written $\mathrm{deg} p = n$.

We said that a number $\lambda$ is a root of the polynomial $p$ if $p(\lambda) = 0$.

We then saw that $\lambda \in \mathbb{F}$ is a root of the polynomial $p$ with $\mathrm{deg} p = n ≥ 1$ if and only if there exists a polynomial $q$ where $\mathrm{deg} q = n - 1$ such that $p$ can be factored as:

(1)

\begin{align} \quad p(x) = (x - \lambda) q(x) \end{align}

From this, we saw that a polynomial $p$ with $\mathrm{deg} p = n$ can have at most $n$ distinct roots.

Lastly, we noted that a polynomial that is the zero function has its coefficients $a_0 = a_1 = ... = a_n = 0$.

We will now look at some examples regarding polynomials.

Example 1

Find all roots to the polynomial $p(x) = x^2 + 3x - 4$.

Note that we can easily factor this quadratic polynomial as $(x + 4)(x - 1) = 0$. Therefore the roots of this polynomial are $x = -4$ and $x = 1$.

Example 2

Determine whether the subset $\{ 0 \} \cup \{ p \in \wp (\mathbb{F}) : \mathrm{deg} p ≤ m \}$ of $\wp ( \mathbb{F})$ is a subspace of $\wp (\mathbb{F})$.

To show that $\{ 0 \} \cup \{ p \in \wp (\mathbb{F}) : \mathrm{deg} p ≤ m\}$ is a subspace of $\wp ( \mathbb{F})$ we must show that the zero polynomial $0 \in \{ 0 \} \cup \{ p \in \wp (\mathbb{F}) : \mathrm{deg} p ≤ m \}$ and that this subset is closed under addition and under scalar multiplication.

Clearly the zero polynomial is contained in this set.

Now let $p(x), q(x) \in \{ 0 \} \cup \{ p \in \wp (\mathbb{F}) : \mathrm{deg} p ≤ m \}$. We have that $\mathrm{deg} (p + q) ≤ \max (\mathrm{deg} p, \mathrm{deg} q)$. Since $p$ and $q$ have degree $-\infty$ or degree $m$ then $\mathrm{deg} (p + q)$ is less than or equal to $m$, so $(p(x) + q(x)) \in \{ 0 \} \cup \{ p \in \wp (\mathbb{F}) : \mathrm{deg} p ≤ m \}$.

Let $a \in \mathbb{F}$ and let $p(x) \in \{ 0 \} \cup \{ p \in \wp (\mathbb{F}) : \mathrm{deg} p ≤ m \}$. Then $ap(x) \in \{ 0 \} \cup \{ p \in \wp (\mathbb{F}) : \mathrm{deg} p ≤ m \}$ since $\mathrm{deg} (ap) = \mathrm{deg} p$.

Therefore $\{ 0 \} \cup \{ p \in \wp (\mathbb{F}) : \mathrm{deg} p ≤ m \}$ is a subspace of $\wp ( \mathbb{F} )$.

Example 3

Let $m$ and $n$ be positive integers where $m ≤ n$. Prove that there exists a polynomial $p$ where $\mathrm{deg} p = n$ and such that $\lambda_1, \lambda_2, ..., \lambda_m \in \mathbb{F}$ are the only roots of $o$.

Consider the following polynomial:

(2)

\begin{align} \quad p(x) = (x - \lambda_1)^{(n - m + 1)}(x - \lambda_2)...(x - \lambda_m) \end{align}

The degree of $(x - \lambda_1)^{(n - m + 1)}$ is $n - m + 1$ and the degree of $(x - \lambda_2)...(x - \lambda_m)$ is $m - 1$, and so $\mathrm{deg} p = n - m + 1 + m - 1 = n$. Furthermore, the only roots of this polynomial are $\lambda_1, \lambda_2, ..., \lambda_m \in \mathbb{F}$. Note that each root has multiplicity $1$ except the root $\lambda_1$ which has multiplicity $n - m + 1$.