Equivalent Representations of Polynomial and Rational Expressions - AP PreCalculus Study Guide

The Core Idea: Equivalent Representations of Polynomial and Rational Expressions

The central concept of this topic is that a single polynomial or rational expression can be written in multiple, algebraically equivalent forms. While these forms look different, they represent the exact same mathematical relationship. The purpose of rewriting expressions is to reveal different properties or to make them more suitable for specific mathematical procedures.

This topic focuses on two primary techniques for creating these equivalent representations. The first is polynomial long division, which is used to rewrite a rational expression $\frac{p(x)}{d(x)}$ as the sum of a polynomial quotient and a simpler rational remainder, $\frac{p(x)}{d(x)}=q(x)+\frac{r(x)}{d(x)}$. The second technique is partial fraction decomposition, which is used to break down a single, complex rational expression into a sum of several simpler fractions, provided the degree of the numerator is less than the degree of the denominator and the denominator is factorable.

Key Rules and Forms

The Polynomial Division Algorithm

When a polynomial $p(x)$ (the dividend) is divided by a non-zero polynomial $d(x)$ (the divisor), the result can be expressed in the form:

$$\frac{p(x)}{d(x)}=q(x)+\frac{r(x)}{d(x)}$$

• $q(x)$ is the quotient polynomial.

• $r(x)$ is the remainder polynomial.

• A critical condition is that the degree of the remainder, $r(x)$, must be strictly less than the degree of the divisor, $d(x)$. If the remainder is $0$, then $d(x)$ is a factor of $p(x)$.

This form is particularly useful for analyzing rational functions where the degree of the numerator is greater than or equal to the degree of the denominator (an "improper" rational function).

Partial Fraction Decomposition Form (Distinct Linear Factors)

For a rational expression where the degree of the numerator is less than the degree of the denominator and the denominator can be factored into distinct (non-repeating) linear factors, the expression can be decomposed into a sum of simpler fractions.

If the denominator factors as $(a_{1}x+b_1)(a_{2}x+b_2)...(a_{n}x+b_n)$, the decomposition will have the form:

$$\frac{p(x)}{(a_{1}x+b_1)(a_{2}x+b_2)...(a_{n}x+b_n)}=\frac{A_1}{a_{1}x+b_1}+\frac{A_2}{a_{2}x+b_2}+...+\frac{A_n}{a_{n}x+b_n}$$

• $A_1,A_2,...,A_n$ are constants that must be determined.

• Each factor in the original denominator gets its own fraction in the sum, with a constant numerator.

Understanding Conditions for Each Technique

A crucial aspect of this topic is knowing when to apply each rewriting technique. The choice is determined by the relationship between the degrees of the numerator and the denominator of the rational expression.

• Use Polynomial Long Division when the degree of the numerator is greater than or equal to the degree of the denominator. The goal is to separate the expression into a polynomial part ($q(x)$) and a "proper" rational part ($\frac{r(x)}{d(x)}$), where the numerator's degree is now less than the denominator's.

• Use Partial Fraction Decomposition when the degree of the numerator is strictly less than the degree of the denominator. This technique does not apply if the degrees are equal or if the numerator's degree is larger. Furthermore, the denominator must be factorable into linear factors for the methods in this course to apply. If an expression is improper, you must perform long division first before you can apply partial fraction decomposition to the resulting remainder term.

Core Concepts & Rules

• A single rational expression can be represented in various equivalent algebraic forms.

• Polynomial long division is the standard algorithm used to divide one polynomial by another.

• The result of dividing a polynomial $p(x)$ by a polynomial $d(x)$ is always expressible as $\frac{p(x)}{d(x)}=q(x)+\frac{r(x)}{d(x)}$.

• In the polynomial division algorithm, the degree of the remainder polynomial $r(x)$ must be less than the degree of the divisor polynomial $d(x)$.

• Partial fraction decomposition is a method for rewriting a single rational expression as a sum of simpler fractions.

• The two conditions for applying partial fraction decomposition are: (1) the degree of the numerator must be less than the degree of the denominator, and (2) the denominator must be factorable.

• If the denominator of a proper rational expression has distinct linear factors, its partial fraction decomposition is a sum of terms of the form $\frac{A}{ax+b}$, where $A$ is a constant.

Step-by-Step Example 1: Polynomial Long Division

Problem: Rewrite the rational expression $f(x)=\frac{3x^3+5x^2-4x-1}{x+2}$ in the form $q(x)+\frac{r(x)}{d(x)}$.

Step 1: Set up the division.

Arrange the polynomials in a long division format. Make sure both polynomials are in standard form (descending powers of $x$) and include placeholders for any missing terms (e.g., $0x^2$).

_________________ x + 2 | 3x^3 + 5x^2 - 4x - 1

Step 2: Divide the leading terms.

Divide the first term of the dividend ($3x^3$) by the first term of the divisor ($x$). The result is $3x^2$. Write this above the division bar.

3x^2 ____________ x + 2 | 3x^3 + 5x^2 - 4x - 1

Step 3: Multiply and subtract.

Multiply the result ($3x^2$) by the entire divisor ($x+2$) to get $3x^2(x+2)=3x^3+6x^2$. Write this result below the dividend and subtract. Be careful with the signs.

3x^2 ____________ x + 2 | 3x^3 + 5x^2 - 4x - 1 -(3x^3 + 6x^2) ___________ -x^2

Step 4: Bring down the next term and repeat.

Bring down the next term from the dividend ($-4x$) to form the new polynomial to be divided. Repeat the process: divide the new leading term ($-x^2$) by the divisor's leading term ($x$) to get $-x$.

3x^2 - x ________ x + 2 | 3x^3 + 5x^2 - 4x - 1 -(3x^3 + 6x^2) ___________ -x^2 - 4x

Step 5: Continue the process.

Multiply $-x$ by the divisor ($x+2$) to get $-x^2-2x$. Subtract this from the current line.

``3x^2 - x ________ x + 2 | 3x^3 + 5x^2 - 4x - 1 -(3x^3 + 6x^2) ___________ -x^2 - 4x -(-x^2 - 2x) __________ -2x`$Bringdownthefinalterm($-1$). Divide $-2x$ by $x$ to get $-2$.

$‘$

    3x^2 -  x - 2 ____

x + 2 | 3x^3 + 5x^2 - 4x - 1

  -(3x^3 + 6x^2)

  ___________

        -x^2 - 4x

      -(-x^2 - 2x)

      __________

             -2x - 1

$‘$

Multiply $-2$ by the divisor ($x+2$) to get $-2x-4$. Subtract this.

$‘$

    3x^2 -  x - 2 ____

x + 2 | 3x^3 + 5x^2 - 4x - 1

  -(3x^3 + 6x^2)

  ___________

        -x^2 - 4x

      -(-x^2 - 2x)

      __________

             -2x - 1

           -(-2x - 4)

           _________

                   3

`` **Step 6: Identify the quotient and remainder.** The process stops because the degree of the result (3, which is degree 0) is less than the degree of the divisor ($x+2, which is degree 1).

• Quotient $q(x)=3x^2-x-2$

• Remainder $r(x)=3$

Final Answer:

The equivalent representation is:

$$\frac{3x^3+5x^2-4x-1}{x+2}=3x^2-x-2+\frac{3}{x+2}$$

Step-by-Step Example 2: Partial Fraction Decomposition

Problem: Find the partial fraction decomposition of the expression $g(x)=\frac{7x+2}{x^2+x-6}$.

Step 1: Check the conditions and factor the denominator.

The degree of the numerator (1) is less than the degree of the denominator (2), so decomposition is appropriate. Now, factor the denominator:

$x^2+x-6=(x+3)(x-2)$.

The factors are distinct and linear.

Step 2: Set up the partial fraction form.

Based on the distinct linear factors, the form of the decomposition will be:

$$\frac{7x+2}{(x+3)(x-2)}=\frac{A}{x+3}+\frac{B}{x-2}$$

where $A$ and $B$ are constants we need to find.

Step 3: Clear the denominators.

Multiply both sides of the equation by the original denominator, $(x+3)(x-2)$:

$$(x+3)(x-2)\left[\frac{7x+2}{(x+3)(x-2)}\right]=(x+3)(x-2)\left[\frac{A}{x+3}+\frac{B}{x-2}\right]$$

This simplifies to:

$$7x+2=A(x-2)+B(x+3)$$

Step 4: Solve for the constants A and B.

We can solve for $A$ and $B$ by substituting convenient values for $x$ that make one of the terms zero.

• To find B, let $x=2$: This will make the $A(x-2)$ term become zero.

  $$7(2)+2=A(2-2)+B(2+3)$$

  $$14+2=A(0)+B(5)$$

  $$16=5B$$

  $$B=\frac{16}{5}$$

• To find A, let $x=-3$: This will make the $B(x+3)$ term become zero.

  $$7(-3)+2=A(-3-2)+B(-3+3)$$

  $$-21+2=A(-5)+B(0)$$

  $$-19=-5A$$

  $$A=\frac{19}{5}$$

Step 5: Write the final decomposed form.

Substitute the values of $A$ and $B$ back into the form from Step 2.

Final Answer:

$$\frac{7x+2}{x^2+x-6}=\frac{19/5}{x+3}+\frac{16/5}{x-2}$$

or, written more cleanly:

$$\frac{7x+2}{x^2+x-6}=\frac{19}{5(x+3)}+\frac{16}{5(x-2)}$`

Using Your Calculator

The procedures in this topic—polynomial long division and partial fraction decomposition—are purely analytical and algebraic. There are no built-in calculator functions to perform these operations directly.

However, a graphing calculator is an excellent tool for checking your answer.

To check your work:

1. Enter the original rational expression into your calculator as Y1.

   • For Example 1: Y1 = (3x^3 + 5x^2 - 4x - 1) / (x + 2)

2. Enter your rewritten, equivalent form into Y2.

   • For Example 1: Y2 = 3x^2 - x - 2 + 3 / (x + 2)

3. Method 1: Graphing. Graph both Y1 and Y2 in the same window. If your algebraic work is correct, the two graphs should be perfectly identical, appearing as a single curve on the screen. If you see two different curves, you have made an error.

4. Method 2: Table of Values. Go to the table feature (2nd + GRAPH). For every value of x` (where the functions are defined), the corresponding values in the `Y1` and `Y2` columns should be exactly the same. If you find any $x-value where Y1 does not equal Y2, your answer is incorrect.

This checking method works for both polynomial long division and partial fraction decomposition results.

AP Exam Quick Hit

Common Question Types

• Finding the Remainder: "What is the remainder when $p(x)=2x^3-5x+1$ is divided by $d(x)=x-3$?" This requires performing polynomial division (or synthetic division, a related shortcut) and identifying only the final remainder.

• Identifying the Correct Decomposition Form: "Which of the following expressions shows the correct form for the partial fraction decomposition of $\frac{4x-1}{x^2-9}$?"

  • (A) $\frac{A}{x^2-9}$

  • (B) $\frac{Ax-1}{x-3}+\frac{B}{x+3}$

  • (C) $\frac{A}{x-3}+\frac{B}{x+3}$

  • (D) $\frac{A}{x}+\frac{B}{x-3}+\frac{C}{x+3}$

  This tests your knowledge of EK 1.11.C2 without requiring calculation.

• Solving for a Single Coefficient: "The expression $\frac{10}{x^2-4}$ can be written in the form $\frac{A}{x-2}+\frac{B}{x+2}$. What is the value of $A$?" This requires you to set up the decomposition and solve for one of the unknown constants.

Common Mistakes

• Sign Errors in Subtraction: The most common error in polynomial long division is failing to distribute the negative sign correctly when subtracting. For example, subtracting $(3x^3+6x^2)$ is equivalent to adding $(-3x^3-6x^2)$.

• Applying Partial Fractions to Improper Expressions: Attempting to use partial fraction decomposition when the degree of the numerator is greater than or equal to the degree of the denominator. You must perform long division first.

• Arithmetic Errors: Simple calculation mistakes when substituting values of $x$ to find the coefficients $A$ and $B$ in partial fraction decomposition.

• Incomplete Answer: After performing long division, stating only the quotient $q(x)$ as the answer and forgetting to include the remainder term $\frac{r(x)}{d(x)}$. The entire expression $q(x)+\frac{r(x)}{d(x)}$ is the equivalent form.

• Incorrectly Factoring the Denominator: An error in factoring the denominator at the beginning of a partial fraction decomposition problem will make the entire rest of the problem incorrect.