Polynomial Functions

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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 non-zero, 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:

  1. 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 \(x \to \infty \), \({a_0}{x^n} \to \infty \).

  2. 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:

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

  2. 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 y-value of negative infinity to a y-value 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.

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

  2. 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.

\[f\left( x \right) = {x^3} - 6{x^2} + 11x - 6\]

Polynomial Functions - Graph 1

Note that the scale of the two axes is different.

The following figure shows the plot of the fifth degree polynomial

\[f\left( x \right) =  - 3{x^5} + 7{x^3} - 10x + 4\]

Polynomial Functions - Graph 2

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

\[f\left( x \right) = {x^4} - 5{x^3} + 2{x^2} - 7x -3\]

Polynomial Functions - Graph 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.

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

  2. 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.

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

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

Thus, f will have a finite positive value. The following figure shows the plot of the sixth-degree polynomial function

\[f\left( x \right) =  - {x^6} + 2{x^4} - {x^2} - 3\]

Polynomial Functions - Graph 4

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.

  1. The graph of any polynomial function has a continuous curve.

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

  3. The range of a polynomial function depends upon its degree:

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

    2. If the degree is even, then

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

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

  4. 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 101-degree 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 \(\mathbb{R}\) .