Question

How do you find the degree, leading term , the leading coefficient , the constant term and end behavior of $g\left( x \right) = 3{x^5} - 2{x^2} + x + 1$ .

Answer 1

Hint: For this type of questions just remember the concept of degree, leading term , the leading coefficient , the constant term and end behavior and their definitions.

Complete step-by-step answer:
The objective of the problem is to find the degree , leading term , the leading coefficient , the constant term and end behavior of $g\left( x \right) = 3{x^5} - 2{x^2} + x + 1$
Given $g\left( x \right) = 3{x^5} - 2{x^2} + x + 1$
Now we are finding the degree of $g\left( x \right) = 3{x^5} - 2{x^2} + x + 1$
Degree: The degree is nothing but the sum of all exponents of all the terms.
Now in the given expression we have 5,2,1 are the exponents.
Now the sum of exponents is 5+2+1=8.
Therefore, the degree of $g\left( x \right) = 3{x^5} - 2{x^2} + x + 1$ is 8.
Leading term: the leading term of polynomial is defined as the term with highest degree.
In the given polynomial $g\left( x \right) = 3{x^5} - 2{x^2} + x + 1$ the term with highest degree is $3{x^5}$ .
Thus, the leading term in $g\left( x \right) = 3{x^5} - 2{x^2} + x + 1$ is $3{x^5}$.
The leading coefficient : The leading coefficient of the polynomial is defined as the number that multiply the leading term.
In the given polynomial the leading term is $3{x^5}$ and the number that multiplies the leading term is 3.
Therefore , the leading coefficient is 3.
The constant term : The constant term of polynomial is defined as the number without the variable.
In the given polynomial the number without the variable is 1.
Therefore, the constant term in $g\left( x \right) = 3{x^5} - 2{x^2} + x + 1$ is 1.
End behavior : The end behavior of the polynomial is behavior of graph as x approaches positive infinity and negative infinity.
The leading term is $3{x^5}$.
Now the end behavior is
$
  \mathop {\lim }\limits_{x \to \infty } 3{x^5} = \infty \\
  \mathop {\lim }\limits_{x \to - \infty } 3{x^5} = - \infty \\
$

Note: To find the end behavior of a polynomial first check whether the function is odd degree or even degree function and also check whether the leading coefficient is positive or negative.