Exact Taylor Polynomials for Polynomial Functions

From Approximation to Exact Representation

A Taylor polynomial is often introduced as a local approximation to a function near a chosen center. For a polynomial, there is an important case in which the approximation is not merely accurate near the center: if the Taylor polynomial includes every degree up to the polynomial’s degree, it reproduces the original polynomial exactly.

The center can be any real number. Changing the center changes the powers and coefficients used to write the polynomial, but it does not change the function represented. The key is to include all terms required by the polynomial’s degree.

Definition and Notation

Let \(p\) be a polynomial and let \(a\) be a real number. The Taylor polynomial of degree \(m\) for \(p\) at \(a\) is formed from the values of \(p\) and its derivatives at \(a\).

The center \(a\) determines the powers \((x-a)^k\). The coefficient of each power is the corresponding derivative value divided by the factorial \(k!\). For instance, the coefficient of \((x-a)^2\) is \(p''(a)/2!\), not simply \(p''(a)\).

If \(p\) has degree \(m\), then its \(m\)-th derivative is constant and every derivative of order greater than \(m\) is zero. The degree-\(m\) Taylor polynomial therefore includes all the potentially nonzero derivative terms.

Why the Centered Form Is Exact

To see why the formula reproduces the polynomial, write the polynomial using powers of \(x-a\). Such a representation is possible because \(x=a+(x-a)\): substituting this into the usual powers of \(x\) and expanding gives a polynomial in \(x-a\). Write that representation as

When this expression is differentiated \(r\) times and evaluated at \(x=a\), the term involving \((x-a)^r\) contributes \(r!b_r\). Terms with a higher power still contain a positive power of \(x-a\) and become zero at \(a\); terms with a lower power have already differentiated to zero. Thus the derivative value identifies exactly the coefficient of the corresponding centered power.

This reasoning gives a complete proof of exactness. Differentiating the centered expression \(r\) times yields

At \(x=a\), the term with \(j=r\) is \(b_r r!\). Every term with \(j>r\) has a positive power of \(x-a\), so its value is zero. Therefore \(p^{(r)}(a)=r!b_r\), and substituting \(b_r=p^{(r)}(a)/r!\) into the centered expression gives the Taylor formula.

The equality holds for all real \(x\), not only for \(x\) close to \(a\). The proof relies on the polynomial having no terms of degree greater than \(m\), and on including every derivative term through order \(m\).

Worked Examples

Degree Matters

The degree of the Taylor polynomial determines how many derivative values are used. For a polynomial of degree \(m\), the degree-\(m\) Taylor polynomial contains every possible nonzero centered coefficient and is exact. If a polynomial has lower degree than the stated Taylor order, its derivatives above its degree are zero; including those zero terms does not change the expression.

By contrast, a Taylor polynomial whose degree is less than the polynomial’s degree usually omits terms from its centered representation. It may still match the function and some of its derivatives at the center, but it need not equal the function for other values of \(x\). The exactness result is not a claim that every low-degree Taylor polynomial is exact; it specifically uses an order at least as large as the polynomial’s degree.

A Common Factorial Mistake

A frequent error is to use derivative values directly as Taylor coefficients, without dividing by factorials. For \(f(x)=x^3\) centered at \(a=1\), the values are \(f(1)=1\), \(f'(1)=3\), \(f''(1)=6\), and \(f'''(1)=6\). Omitting the factorials would give the incorrect expression

At \(x=2\), this expression gives \(1+3+6+6=16\), while \(f(2)=8\). The correct coefficients divide the second- and third-derivative values by \(2!\) and \(3!\):

At \(x=2\), this gives \(1+3+3+1=8\), as required. Expanding the corrected expression gives \((1+(x-1))^3=x^3\), confirming the exact identity.

Why Exact Centered Forms Are Useful

A centered form displays the polynomial relative to a chosen point. At \(x=a\), every positive power of \(x-a\) vanishes, so the constant term immediately gives \(p(a)\). The coefficient of \(x-a\) records the slope at the center, and higher coefficients record the corresponding higher derivatives divided by factorials.

For a polynomial of matching degree, this information is enough to reconstruct the entire function, not just its behavior near the center. The expression may be more convenient when calculations or interpretation focus on a particular value of \(x\), while the ordinary powers of \(x\) may be more convenient elsewhere. Both forms represent the same polynomial.