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\(a(x)\) \(b(x)\) \(q(x)+\tfrac{r(x)}{b(x)}\)

Rewriting
Rational Expressions

Learn to transform \(\dfrac{a(x)}{b(x)}\) into the form \(q(x) + \dfrac{r(x)}{b(x)}\) using inspection, long division, and algebraic reasoning.

Algebra 2

What Is a Rational Expression?

A rational expression is any expression that can be written as a ratio of two polynomials:

Numerator \(a(x)\)
The polynomial on top: could be any degree.
Denominator \(b(x)\)
The polynomial on bottom: must not equal zero.

Why Rewrite Rational Expressions?

Just like dividing \(\frac{17}{5} = 3 + \frac{2}{5}\) gives a quotient and remainder, we can decompose rational expressions to reveal hidden structure.

1
Graphing: Identifies the slant (oblique) asymptote \(y = q(x)\).
2
Calculus prep: Makes integration and differentiation much easier.
3
End behavior: Shows how the function behaves as \(x \to \pm\infty\).

The Integer Analogy

You already know this pattern from arithmetic. Polynomial division works the same way:

Integers
$$\frac{17}{5} = 3 + \frac{2}{5}$$
Polynomials
$$\frac{a(x)}{b(x)} = q(x) + \frac{r(x)}{b(x)}$$

The Division Algorithm for Polynomials

\(a(x)\)
Dividend: the numerator polynomial
\(b(x)\)
Divisor: the denominator polynomial
\(q(x)\)
Quotient: the polynomial part
\(r(x)\)
Remainder: degree less than \(b(x)\)

Visualizing the Decomposition

\(a(x)\) \(b(x)\) divide \(q(x)\) Quotient + \(r(x)\) \(b(x)\) Remainder fraction \(\deg(r) < \deg(b)\) Original Rewritten Form

Inspection vs. Long Division

Inspection Long Division
When to use Denominator is linear or simple Any polynomials, any degrees
Speed Fast: mental math Slower: systematic steps
Example \(\dfrac{2x+5}{x+1}\) \(\dfrac{x^3-2x+4}{x^2+1}\)
Key idea Rewrite numerator to contain denominator Divide leading terms, multiply, subtract, repeat

Method 1: Rewriting by Inspection

The idea: manipulate the numerator so that part of it is exactly divisible by the denominator, leaving a known remainder.

1
Identify the denominator \(b(x)\).
2
Rewrite the numerator \(a(x)\) by adding and subtracting a constant so that part of it is a multiple of \(b(x)\).
3
Split the fraction into two pieces: the quotient plus the remainder over the denominator.

Example: \(\dfrac{2x+5}{x+1}\) by Inspection

1
We need the numerator to contain \((x+1)\). Notice \(2x + 5 = 2(x+1) + 3\).
2
Split: \(\dfrac{2(x+1) + 3}{x+1} = \dfrac{2(x+1)}{x+1} + \dfrac{3}{x+1}\).
3
Simplify: \(\dfrac{2(x+1)}{x+1} = 2\).

Here: \(q(x) = 2\), \(r(x) = 3\), and \(\deg(r) = 0 < 1 = \deg(b)\). ✓

Method 2: Polynomial Long Division

For more complex expressions, use the systematic algorithm that mirrors integer long division:

1
Divide the leading term of the dividend by the leading term of the divisor.
2
Multiply the entire divisor by that result.
3
Subtract to get a new (reduced) dividend.
4
Repeat until the degree of the remainder is less than the degree of the divisor.

Example: \(\dfrac{x^2 + 3x + 5}{x + 1}\)

1
Divide: \(\dfrac{x^2}{x} = x\). Write \(x\) in the quotient.
2
Multiply: \(x \cdot (x+1) = x^2 + x\). Subtract: \((x^2+3x+5) - (x^2+x) = 2x + 5\).
3
Divide: \(\dfrac{2x}{x} = 2\). Write \(+2\) in the quotient.
4
Multiply: \(2 \cdot (x+1) = 2x + 2\). Subtract: \((2x+5)-(2x+2) = 3\).

Example: \(\dfrac{x^3 - 2x + 4}{x^2 + 1}\)

1
Divide: \(\dfrac{x^3}{x^2} = x\). Multiply: \(x(x^2+1)=x^3+x\).
2
Subtract: \((x^3 - 2x + 4) - (x^3 + x) = -3x + 4\).
3
Stop: \(\deg(-3x+4) = 1 < 2 = \deg(x^2+1)\). Done!

The quotient is \(q(x) = x\) (the slant asymptote!) and the remainder is \(r(x) = -3x + 4\).

Graphical Interpretation

x y 1 2 3 4 5 6 -1 -2 -3 -4 1 2 3 4 -1 -2 -3 -4 \(y = x\) \(f(x)\) As \(x\) grows, \(f(x)\) approaches \(y = x\) \(\frac{r(x)}{b(x)}\) vanishes

Always Verify Your Result

After dividing, check your work by multiplying back. If correct:

Let's verify our second example:

1
We found: \(\dfrac{x^2+3x+5}{x+1} = x+2 + \dfrac{3}{x+1}\).
2
Check: \((x+2)(x+1) + 3 = x^2 + 3x + 2 + 3 = x^2 + 3x + 5\). ✓

Common Mistakes to Avoid

✕ Forgetting Placeholder Terms
If the dividend is \(x^3 + 4\), write it as \(x^3 + 0x^2 + 0x + 4\) before dividing. Missing terms cause alignment errors.
✕ Sign Errors in Subtraction
When subtracting, distribute the negative sign to every term. \((2x+5) - (2x+2) = 3\), not \(7\).
✕ Stopping Too Early or Late
Stop when \(\deg(r) < \deg(b)\). Don't keep dividing once the remainder's degree is already smaller.

Summary: Choosing Your Method

Situation Best Method Result Form
Linear denominator, simple numerator Inspection \(c + \frac{r}{x+a}\)
Quadratic or higher denominator Long Division \(q(x) + \frac{r(x)}{b(x)}\)
Very complex polynomials CAS / Technology Exact symbolic output

Question 1 of 5

Rewrite \(\dfrac{3x + 7}{x + 2}\) in the form \(q + \dfrac{r}{x+2}\). What are \(q\) and \(r\)?

Question 2 of 5

What is the quotient \(q(x)\) when you divide \(x^2 - 4x + 7\) by \(x - 3\)?

Question 3 of 5

When rewriting \(\dfrac{a(x)}{b(x)} = q(x) + \dfrac{r(x)}{b(x)}\), which condition must the remainder satisfy?

Question 4 of 5

Divide \(2x^3 + 1\) by \(x^2 - 1\). What is the remainder \(r(x)\)?

Question 5 of 5

If \(\dfrac{x^2 + 5x + 8}{x + 2} = x + 3 + \dfrac{r}{x+2}\), what is the value of \(r\)?