Question

Divide: $6{m^2} - 16{m^3} + 10{m^4}$ by $ - 2m$.

Answer 1

Hint: In this problem, we have to divide quartic polynomial (a polynomial of degree $4$) by linear polynomial (a polynomial of degree $1$). For this, first we will simplify the quartic polynomial $6{m^2} - 16{m^3} + 10{m^4}$. Then, we will divide by linear polynomial $ - 2m$ to find the required answer.

Complete step-by-step answer:
In this problem, we have a quartic polynomial is $6{m^2} - 16{m^3} + 10{m^4}$. Rearrange the terms, we get $10{m^4} - 16{m^3} + 6{m^2}$. Let us say $p\left( m \right) = 10{m^4} - 16{m^3} + 6{m^2}$. Let us simplify $p\left( m \right)$ by taking $2m$ common out. So, we can write $p\left( m \right) = 2m\left( {5{m^3} - 8{m^2} + 3m} \right)$. Also we have linear polynomials $ - 2m$. Let us say $q\left( m \right) = - 2m$. Now we are going to divide $p\left( m \right)$ by $q\left( m \right)$. So, we can write
$\dfrac{{p\left( m \right)}}{{q\left( m \right)}} = \dfrac{{2m\left( {5{m^3} - 8{m^2} + 3m} \right)}}{{ - 2m}}$
By cancelling the same factor $2m$ from numerator and denominator, we get
$
  \dfrac{{p\left( m \right)}}{{q\left( m \right)}} = - \left( {5{m^3} - 8{m^2} + 3m} \right) \\
   \Rightarrow \dfrac{{p\left( m \right)}}{{q\left( m \right)}} = - 5{m^3} + 8{m^2} - 3m \\
 $
Therefore, we can say that if we divide $6{m^2} - 16{m^3} + 10{m^4}$ by $ - 2m$ then we get $ - 5{m^3} + 8{m^2} - 3m$ which is required answer.

Note: In this problem, also we can use a long division method to find required division (answer). Also we can factorize $ - 5{m^3} + 8{m^2} - 3m$ by splitting the middle term. Let us take $ - m$ common out. So, we get $ - m\left( {5{m^2} - 8m + 3} \right)$. Now we will split the middle term by finding two numbers whose sum is $ - 8$ and product is $15$. So, we can write
$
   - 5{m^3} + 8{m^2} - 3m \\
   = - m\left( {5{m^2} - 5m - 3m + 3} \right) \\
   = - m\left[ {5m\left( {m - 1} \right) - 3\left( {m - 1} \right)} \right] \\
   = - m\left( {5m - 3} \right)\left( {m - 1} \right) \\
 $
This is also a required answer to a given problem. The polynomial of degree $1$ is called linear polynomial and the polynomial of degree $4$ is called quartic polynomial. Every quartic polynomial has exactly four 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.