Use Synthetic Division to Analyze Rational End Behavior

The quotient from synthetic division is the polynomial asymptote of a rational function when the divisor is a monic linear factor of the denominator. For P(x) = 2x³ + 3x² - 4x + 5 divided by x - 2, the quotient 2x² + 7x + 10 is the curvilinear asymptote as x → ∞. This use of synthetic division to analyze rational end behavior works only when the divisor is of the form x - c. The remainder 25/(x - 2) vanishes at infinity, so the quotient alone describes the end behavior. Here is exactly how a linear division result identifies the quotient related to an asymptote.

Take the rational function R(x) = (2x³ + 3x² - 4x + 5)/(x - 2). The denominator is x - 2, so c = 2. Set up the synthetic division table with the coefficients of the dividend: 2, 3, -4, 5. Bring down the leading coefficient 2. Multiply 2 by c = 2 to get 4, add to the next coefficient 3 to get 7. Multiply 7 by 2 to get 14, add to -4 to get 10. Multiply 10 by 2 to get 20, add to 5 to get 25. The bottom row reads 2, 7, 10, 25. The first three numbers are the quotient coefficients: 2x² + 7x + 10. The last number, 25, is the remainder. The quotient 2x² + 7x + 10 is the polynomial part of R(x) that describes the end behavior. As x → ∞, R(x) behaves like 2x² + 7x + 10 because the remainder term 25/(x - 2) approaches zero.

Perform the synthetic division calculation step by step for the example. Write the coefficients of the dividend in a row: 2, 3, -4, 5. The divisor x - 2 gives c = 2. Draw a horizontal line. Bring down the leading coefficient 2 below the line. Multiply 2 by c = 2 to get 4. Place this product under the next coefficient 3 and add: 3 + 4 = 7. Write 7 below the line. Multiply 7 by c = 2 to get 14. Add to -4: -4 + 14 = 10. Write 10 below the line. Multiply 10 by c = 2 to get 20. Add to 5: 5 + 20 = 25. Write 25 below the line. The bottom row is 2, 7, 10, 25. The quotient coefficients are 2, 7, 10, forming Q(x) = 2x² + 7x + 10. The remainder is 25. The degree of the quotient, 2, is one less than the degree of the dividend, 3. This is always true for a linear divisor. The quotient dimension is coefficients; the asymptote is a polynomial function.

The Remainder Theorem states that P(c) equals the remainder, which is 25 in this example. The Factor Theorem says that if the remainder were zero, x - c would be a factor. For asymptote analysis, the quotient is the key, not the remainder. The rational function R(x) = P(x)/(x - c) can be written as R(x) = Q(x) + R/(x - c). As x → ∞, the term R/(x - c) tends to zero, so R(x) approaches Q(x). This is the oblique or curvilinear asymptote. The asymptote is an authentic use of the quotient when the divisor qualifies as a monic linear binomial. The quotient from synthetic division is the polynomial asymptote that describes the end behavior of the rational function. This method only works for linear divisors of the form x - c. For divisors like 2x - 3, an extra step is needed: divide the quotient by the leading coefficient of the divisor. For quadratic or higher-degree divisors, use polynomial long division.

The quotient from synthetic division is the polynomial part of the asymptote. When you divide P(x) by x - c, the result is P(x) = (x - c)Q(x) + R. For the rational function P(x)/(x - c), this becomes Q(x) + R/(x - c). The term R/(x - c) is the remainder term. As x → ∞, R/(x - c) → 0. Therefore, the end behavior of the rational function is exactly Q(x). This is why the quotient identifies the asymptote. The remainder term vanishes, leaving the quotient as the sole descriptor of behavior at large x. This holds for any monic linear divisor. The quotient coefficients are dimensionless, and the asymptote is a polynomial function of degree one less than the dividend.

Synthetic division as described only works for divisors of the form x - c. If the divisor is 2x - 3, the standard algorithm does not apply directly. The divisor 2x - 3 must be rewritten as 2(x - 3/2). Dividing P(x) by 2x - 3 requires first dividing by x - 3/2 using synthetic division, then dividing the resulting quotient by 2. This extra step is often omitted in textbooks. For example, dividing P(x) = 2x³ + 3x² - 4x + 5 by 2x - 3, set c = 3/2. Perform synthetic division with c = 1.5. The bottom row gives quotient coefficients for a divisor of x - 1.5. Then divide each quotient coefficient by 2 to get the true quotient for 2x - 3. Skipping this step yields a quotient off by a factor of 2. This is the most common error when using synthetic division for non-monic divisors. Use the asymptote calculator to verify your result.