Question

How do you use the leading coefficient test to determine the end behaviour of the polynomial function $f\left( x \right) = - 5\left( {{x^2} + 1} \right)\left( {x - 2} \right)$?

Answer 1

Hint: Leading coefficient is the coefficient of the first variable or the variable with the highest power attached. For this, first, simplify the equation and then arrange it in ascending order or in the order of highest power to lowest power. Thereafter observe the first variable or the leading variable to determine the nature of the function.

Complete step by step solution:
For this question, what we need to do is first simplify the given equation and expand it.
$\
  f\left( x \right) = - 5\left( {{x^2} + 1} \right)\left( {x - 2} \right) \\
   = - 5\left[ {x\left( {{x^2} + 1} \right) - 2\left( {{x^2} + 1} \right)} \right] \\
   = - 5\left[ {{x^3} + x - 2{x^2} - 2} \right] \\
   = - 5\left[ {{x^3} - 2{x^2} + x - 2} \right] \\
   = - 5{x^3} + 10{x^2} - 5x + 10 \\
\ $
Here the variable with the highest power shall be observed along with its coefficient which is $ - 5{x^3}$. Now what happens is that we need to look at the power of this variable. If it is even, its graph will show the same behavior on both sides of the y axis but if it is odd, its behavior will change with the change inside. What this means, we will understand in a bit.
Before that, we need to observe the -5 attached here. This shows that when we move to the positive side of x, this will make the overall $ - 5{x^3}$ negative which in turn will lead to the whole function to decrease in y-axis at the right half and when we put negative values, it will make the function increase in the left.
As such we can say that this function increases on the left and decreases on the right side of the graph. Or in other notation, we can say that it rises on the left and falls on the right side of the graph as per the leading variable.
Now back to the even leading variables as mentioned above. If the leading variable has even power, it will either rise on both sides or fall on both sides depending on its coefficient.

Note:
The leading variable can only show how the function will behave and can never give the location of its roots. It helps only in determining what will be the sign of a value if put in the function and that value also needs to be very large, so large that it can never fall in between the roots of the equation.