Question

Divide $\left( {{m^2} - 14m - 32} \right)$ by $\left( {m + 2} \right)$.

Answer 1

Hint: In this problem, we have to divide quadratic polynomial (a polynomial of degree $2$) by linear polynomial (a polynomial of degree $1$). For this, first we will factorize the quadratic polynomial ${m^2} - 14m - 32$. Then, we will divide by linear polynomial $m + 2$ to find the required answer.

Complete step-by-step answer:
In this problem, we have a quadratic polynomial which is given as ${m^2} - 14m - 32$. Let us say $p\left( m \right) = {m^2} - 14m - 32$. Let us factorize $p\left( m \right)$ by splitting the middle term. We have to find two numbers whose product is $ - 32$ and sum is $ - 14$. These numbers are $ - 16$ and $2$ because products of $ - 16$ and $2$ is $ - 32$. Also the sum of $ - 16$ and $2$ is $ - 14$. So, we can write
$
  p\left( m \right) = {m^2} - 16m + 2m - 32 \\
   \Rightarrow p\left( m \right) = m\left( {m - 16} \right) + 2\left( {m - 16} \right) \\
   \Rightarrow P\left( m \right) = \left( {m + 2} \right)\left( {m - 16} \right) \\
 $
Also we have a linear polynomial is $m + 2$. Let us say $q\left( m \right) = m + 2$. Now we are going to divide $p\left( m \right)$ by $q\left( m \right)$. So, we can write
$
  \dfrac{{p\left( m \right)}}{{q\left( m \right)}} = \dfrac{{{m^2} - 14m - 32}}{{m + 2}} \\
   \Rightarrow \dfrac{{p\left( m \right)}}{{q\left( m \right)}} = \dfrac{{\left( {m + 2} \right)\left( {m - 16} \right)}}{{m + 2}} \\
 $
By cancelling the same factor $\left( {m + 2} \right)$ from numerator and denominator, we get
$\dfrac{{p\left( m \right)}}{{q\left( m \right)}} = m - 16$
Hence, we can say that if we divide $\left( {{m^2} - 14m - 32} \right)$ by $\left( {m + 2} \right)$ then we get $\left( {m - 16} \right)$ which is required answer.

Note: In this problem, we can also use a long division method to find required division (answer). The highest power of $m$ in the polynomial $\left( {{m^2} - 14m - 32} \right)$ is called the degree of this polynomial. The polynomial of degree $1$ is called linear polynomial and the polynomial of degree $2$ is called quadratic polynomial. Every quadratic polynomial has exactly two factors. Remember that a polynomial of one term is called a monomial, a polynomial of two terms is called a binomial and a polynomial of three terms is called a trinomial.