Polynomial Functions
We have already discussed polynomials in detail. A polynomial function of degree n has the following definition:
\[f\left( x \right) = {a_0}{x^n} + {a_1}{x^{n  1}} +{a_2}{x^{n  2}} + ... + {a_{n  1}}x + {a_n}\]
The coefficient of the highest degree term should be nonzero, otherwise f will be a polynomial of a lower degree.
All polynomial functions are defined over the set of all real numbers. In other words, the domain of any polynomial function is \(\mathbb{R}\).
The range of a polynomial function depends on the degree of the polynomial. Let us see how. Consider the polynomial
\[f\left( x \right) = {a_0}{x^n} + {a_1}{x^{n  1}} +{a_2}{x^{n  2}} + ... + {a_{n  1}}x + {a_n}\]
once again. When x takes on very large values, the highest degree term \({a_0}{x^n}\)dominates in magnitude compared to the other terms. For example, let
\[f\left( x \right) = {x^5} + {x^4} + 1\]
Take a really large value for x,say \(x = {10^6}\). For this value of x,the highest degree term has a magnitude of \({10^{30}}\),while the second term has a magnitude of (only) \({10^{24}}\). The first term is larger than the second term by a factor of \({10^6}\) or 1 million. Clearly, for really large values of x, only the highest degree term matters – the other terms have relatively negligible magnitudes.
Now, suppose that the degree of the \(f\left( x \right)\) polynomial is odd. Also,suppose that the value of \({a_0}\) in the highest degree term \({a_0}{x^n}\) is positive.
We make the following observations:

When x takes on large positive values, then \({a_0}{x^n}\)also takes on large positive values. We can make \({a_0}{x^n}\) as large as possible by taking a sufficiently large value of x. Thus, we can say that as x tends to infinity, the term \({a_0}{x^n}\) also tends to infinity: as \(x \to \infty \), \({a_0}{x^n} \to \infty \).

When x takes on large negative values, \({a_0}{x^n}\)also takes on large negative values, because the power of x in this term(which is n) is odd. That is: as \(x \to  \infty \), \({a_0}{x^n} \to  \infty \).
Since the value of f for large values of x is mostly influenced by the highest degree term, we can say that:

As \(x \to \infty \), \(f \to \infty \).

As \(x \to  \infty \), \(f \to  \infty \).
Also, any polynomial function has a continuous variation, as mentioned earlier (there are no breaks in the graph).This means that the curve for f will vary (continuously) from a yvalue of negative infinity to a yvalue of positive infinity. In other words,the range of f is \(\mathbb{R}\).
If the coefficient \({a_0}\) in the highest degree term \({a_0}{x^n}\) is negative, then.

As \(x \to \infty \), \(f \to  \infty \).

As \(x \to  \infty \), \(f \to \infty \).
Once again, f will have a range of \(\mathbb{R}\).
We conclude that for any polynomial function with an odd degree, the range will be \(\mathbb{R}\) . Let us see some examples.
The following figure shows the plot of.
\[f\left( x \right) = {x^3}  6{x^2} + 11x  6\]
Note that the scale of the two axes is different.
The following figure shows the plot of the fifth degree polynomial
\[f\left( x \right) =  3{x^5} + 7{x^3}  10x + 4\]
In both cases, it is easy to see that the range of the polynomial function is \(\mathbb{R}\).
What can we say about the range of a polynomial function of even degree? We have already seen that the curve for a quadratic function is a parabola opening upward or downward, depending upon whether the coefficient of the square term is positive or negative(respectively). This means that the range of a quadratic function is not \(\mathbb{R}\) but a proper subset of \(\mathbb{R}\).
The following figure shows the plot of the fourth degree polynomial function
\[f\left( x \right) = {x^4}  5{x^3} + 2{x^2}  7x 3\]
Both the arms of the curve tend towards infinity. On the negative side, there is a certain value below which the curve does not go. The range of this function is therefore a subset of \(\mathbb{R}\)  from the minimum value to positive infinity.
Why do polynomial functions with even degrees don’t have ranges of \(\mathbb{R}\)? If the coefficient \({a_0}\) in the highest degree term \({a_0}{x^n}\) is positive,then f will tend to positive infinity on both sides. That is.

As \(x \to \infty \), \(f \to \infty \).

As \(x \to  \infty \), \(f \to \infty \).
Thus, the curve never goes towards negative infinity. It has a finite minimum value.
On the other hand, if the coefficient \({a_0}\) in the highest degree term \({a_0}{x^n}\) is negative, then f will tend to negative infinity on both sides.

As\(x \to \infty \), \(f \to  \infty \).

As\(x \to  \infty \), \(f \to  \infty \).
Thus, f will have a finite positive value. The following figure shows the plot of the sixthdegree polynomial function
\[f\left( x \right) =  {x^6} + 2{x^4}  {x^2}  3\]
Clearly, f has a finite maximum value (somewhere between 20 and 40). The range of f will be negative infinity to this maximum value.
Let us now summarize our discussion on polynomial functions.

The graph of any polynomial function has a continuous curve.

The domain of any polynomial function is \(\mathbb{R}\).

The range of a polynomial function depends upon its degree:

If the degree is odd, then the range is \(\mathbb{R}\).

If the degree is even, then

If the coefficient of the highest degree term is positive, the range will be a certain minimum value to positive infinity.

If the coefficient of the highest degree term is negative, the range will be negative infinity to a certain maximum value.



A polynomial function will assume every value in its range, as the curve for the function is continuous.
Example 1: What can you say about the range of a 101degree polynomial? Will it be \(\mathbb{R}\) , or a subset of \(\mathbb{R}\) ?.
Solution: Since the degree of the polynomial is odd, its range will be \(\mathbb{R}\) .