1.11 Equivalent Representations of Polynomial and Rational Expressions

Rewriting reveals structure

Polynomial and rational expressions can be rewritten in equivalent forms to reveal different information. Factoring shows zeros and multiplicities; expanding shows end behavior via the leading term. Rewriting doesn't change the function's value, just how easily we can read its properties.

Polynomial long division

Polynomial long division rewrites a rational function as a quotient plus remainder over divisor: f(x) = q(x) + r(x)/d(x). This reveals slant/curved asymptotes (from q(x)) and helps analyze behavior as x approaches infinity, since the remainder term shrinks toward zero for large x.

Choosing the useful form

Factored form exposes zeros, holes, and vertical asymptotes. Expanded (standard) form exposes the leading coefficient and degree, useful for end behavior. Division form separates a function into a polynomial part plus a decaying remainder, useful for identifying asymptotic behavior far from removable discontinuities.

Key terms

Factored form (use)

Reveals zeros, holes, and vertical asymptotes of a polynomial or rational function.

Expanded/standard form (use)

Reveals the leading coefficient and degree, useful for determining end behavior.

Polynomial long division result

Rewrites f(x) = n(x)/d(x) as q(x) + r(x)/d(x), a quotient plus remainder over the divisor.

Why the remainder term matters less as x→∞

Since degree(r) < degree(d), r(x)/d(x) → 0 as x grows large, so f(x) behaves like q(x).

Equivalent expressions

Different-looking expressions that produce the same output for every input in the domain, revealing different structural information.

Try an AP-style question

Which is the result of (x² + 3x + 5) ÷ (x + 1)?

1. x + 2 + 3/(x + 1)
2. x + 3 + 5/(x + 1)
3. x + 2 + 5/(x + 1)
4. x + 4 + 1/(x + 1)