Question

How do you factor ${{x}^{2}}+4x+1$?

Answer 1

Hint: The polynomial given to us in the question is a two degree polynomial. So we can use the middle term technique for factoring it. But for this polynomial, the middle term technique will not work. So we have to find out the roots of the given polynomial by equating the polynomial to zero. For this we need to use the quadratic formula, which is given as $x=\dfrac{-b\pm \sqrt{D}}{2a}$, where $D$ is the discriminant. The discriminant $D$ is given by the formula $D={{b}^{2}}-4ac$. In these formulae, $a$ is the coefficient of ${{x}^{2}}$, $b$ is the coefficient of $x$, and $c$ is the constant term in the given quadratic equation. As is clear from the quadratic formula, we will obtain two roots of the polynomial. Finally using the factor theorem, we will be able to factorise the given polynomial from these roots.

Complete step by step answer:
Let us represent the polynomial given in the question as
$p\left( x \right)={{x}^{2}}+4x+1.........(i)$
Since the highest power of the variable $x$ in the given polynomial is $2$, so this is a quadratic polynomial in $x$.
Now, we see that the middle term of the given equation is equal to $4x$, the first term is equal to ${{x}^{2}}$, and the third term is equal to $1$.
By the middle term splitting method, we must split the term $4x$ into the sum of two terms such that their product is equal to the product of the first and the third terms, that is equal to ${{x}^{2}}$. But there is no way to do that. So we cannot use the middle term technique here.
So we try to obtain the roots of the above polynomial by equating it to zero, that is,
$\Rightarrow p\left( x \right)=0$
From (i)
$\Rightarrow {{x}^{2}}+4x+1=0........(ii)$
So we have obtained a quadratic equation in the variable $x$.
We know that the solution of the quadratic equation is given by the quadratic formula which is given as
$x=\dfrac{-b\pm \sqrt{D}}{2a}.......(iii)$
So we need to calculate the discriminant of the given quadratic equation. We know that the discriminant is given by
$D={{b}^{2}}-4ac.......(iv)$
From the equation (ii) we note the following values of the coefficients
$\begin{align}
  & \Rightarrow a=1........(v) \\
 & \Rightarrow b=4........(vi) \\
 & \Rightarrow c=1........(vii) \\
\end{align}$
Substituting (v), (vi) and (vii) in (iv), we get
$\Rightarrow D={{\left( 4 \right)}^{2}}-4\left( 1 \right)\left( 1 \right)$
$\Rightarrow D=16-4$
On solving we get
$\Rightarrow D=12.......(viii)$
Now, we substitute (v), (vi) and (viii) in the quadratic formula given in the equation (iii) to get
$\Rightarrow x=\dfrac{-4\pm \sqrt{12}}{2\left( 1 \right)}$
We know that $\sqrt{12}=2\sqrt{3}$. Substituting this above, we get
$\begin{align}
  & \Rightarrow x=\dfrac{-4\pm 2\sqrt{3}}{2\left( 1 \right)} \\
 & \Rightarrow x=\dfrac{-4+2\sqrt{3}}{2\left( 1 \right)}\text{, }x=\dfrac{-4-2\sqrt{3}}{2\left( 1 \right)} \\
\end{align}$
On solving, we get the roots of the given polynomial as
\[\Rightarrow x=-2+\sqrt{3}\text{, }x=-2-\sqrt{3}\]
By the factor theorem we know that if $k$ is a root of the polynomial $f\left( x \right)$, then $\left( x-k \right)$ is a factor of $f\left( x \right)$.
Since \[x=-2+\sqrt{3}\text{, }x=-2-\sqrt{3}\] are the roots of the given polynomial $p\left( x \right)$, so by factor theorem \[\left( x-\left( -2+\sqrt{3} \right) \right)\] and \[\left( x-\left( -2-\sqrt{3} \right) \right)\] are the factors of $p\left( x \right)$.
Hence, the given polynomial $p\left( x \right)$ can be factored as
\[\begin{align}
  & \Rightarrow p\left( x \right)=\left( x-\left( -2+\sqrt{3} \right) \right)\left( x-\left( -2-\sqrt{3} \right) \right) \\
 & \Rightarrow p\left( x \right)=\left( x+2-\sqrt{3} \right)\left( x+2+\sqrt{3} \right) \\
\end{align}\]

Note:
For factoring a quadratic polynomial, techniques such as middle term splitting, and hit and trial method are very commonly used. But these techniques do not work if the roots of the quadratic polynomial are not rational. So before using these techniques it is advisable to check for the value of the discriminant of the quadratic equation obtained after equating the given quadratic polynomial to zero.