Wemazi

Topic 6.12 · BC

Prepare the rational expression

Builds on using endpoint and integral data. 7 minutes.

A fraction that follows a line

The curve shows the fraction \(\dfrac{x^3 - 2x^2 + 7}{x^2 - x - 6}\). The degree of a polynomial is its highest power of \(x\), so the numerator has degree 3 and the denominator has degree 2.

The dashed vertical lines are at \(x = 3\) and \(x = -2\), where the denominator is 0, and the curve does not cross them.

The faint line is \(y = x - 1\). Drag the orange handle on the x-axis to the right. The orange gap between the curve and the line becomes smaller.

Far from the dashed lines, the fraction is almost the line \(x - 1\). Long division finds this line, the polynomial part, and the gap is the remainder fraction, which shrinks toward 0.

Any fraction of polynomials

A fraction \(P(x)/Q(x)\) of two polynomials splits into a polynomial \(q(x)\), the quotient, plus a remainder fraction.

The faint line shows the quotient, and the dotted curve shows the remainder fraction. When you move a slider, the line and the chalk curve move together, because the chalk curve is their sum.

The dotted curve and the dashed lines stay in place, because the sliders change only the quotient. The numerator of the remainder fraction has a smaller degree than its denominator. Only such a fraction splits into partial fractions, simple fractions with one linear factor in each denominator.

The dashed lines stand at the roots of the denominator, the values of x that make it 0, so its factors locate them.

Notation

  1. A rational expression is a fraction \(\dfrac{P(x)}{Q(x)}\) of two polynomials. The fraction is proper when \(P\) has a smaller degree than \(Q\).

  2. If the fraction is not proper, divide \(P\) by \(Q\) with long division. The result is a quotient \(\cq{q}\) and a remainder \(R\).

    \[\frac{P(x)}{Q(x)} = \cq{q(x)} + \frac{R(x)}{Q(x)}\]

    The remainder \(R\) has a smaller degree than \(Q\), so the new fraction is proper.

  3. Factor \(Q\) into linear factors, which have the form \(x - r\) or \(ax + b\). Take out any common factor first.

    \[x^2 - x - 6 = (x - 3)(x + 2)\]

    Each factor is 0 at one root, and the dashed lines stand at these roots.

  4. Test the factors. Multiply them to find \(Q\) again, or substitute each root into \(Q\) to find 0.

  5. Only then split the proper fraction into partial fractions, the two faint curves, one over each factor. The next pages show how.

When it fails

The denominator \(x^2 + 4\) is never 0, because \(x^2\) is not negative. It has no real roots, so it has no linear factors.

The curve has no dashed lines, so partial fractions with linear factors do not apply here.

Example 1: Factor the denominator

Prepare \(\dfrac{x + 4}{2x^2 - 5x - 3}\) for partial fractions.

  1. The numerator has degree 1 and the denominator has degree 2, so the fraction is proper and needs no division. Far to the left and right, the curve is close to the x-axis, because the denominator grows faster than the numerator.

  2. Factor the denominator. Find two numbers with product \(2\cdot(-3) = -6\) and sum \(-5\), which are \(-6\) and \(1\). Use them to split the middle term.

    \[\begin{aligned} &2x^2 - 5x - 3 \\ &\quad = 2x^2 - 6x + x - 3 \\ &\quad = 2x(x - 3) + (x - 3) \\ &\quad = (2x + 1)(x - 3) \end{aligned}\]

  3. Test each root. Both \(x = 3\) and \(x = -\tfrac12\) make the denominator 0, and they match the dashed lines.

    \[\begin{gathered} 2\cdot 3^2 - 5\cdot 3 - 3 = 0 \\ 2\left(-\tfrac12\right)^2 - 5\left(-\tfrac12\right) - 3 = 0 \end{gathered}\]

Example 2: Divide first

Prepare \(\dfrac{2x^3 + x^2 - 4x + 5}{x^2 - 4}\) for partial fractions.

  1. The numerator has degree 3 and the denominator has degree 2, so the fraction is not proper.

  2. A tempting wrong move: Factor the denominator and split the fraction at once, with constants \(A\) and \(B\).

    \[\begin{aligned} &\frac{2x^3 + x^2 - 4x + 5}{(x - 2)(x + 2)} \\ &\quad = \frac{A}{x - 2} + \frac{B}{x + 2} \end{aligned}\]

    Far from the dashed lines, the chalk curve rises along the faint line \(y = 2x + 1\). The red curve is the right side for one choice of \(A\) and \(B\). Far away it comes close to the x-axis, because each piece does, so no \(A\) and \(B\) can follow the chalk curve.

  3. Divide first. Long division gives the quotient \(2x + 1\), the faint line, and the remainder \(4x + 9\). To test this, multiply the quotient by the denominator and add the remainder.

    \[\begin{aligned} &2x^3 + x^2 - 4x + 5 \\ &\quad = (2x + 1)(x^2 - 4) + (4x + 9) \end{aligned}\]

  4. The dotted curve is the remainder fraction. Its denominator is \(x^2 - 4 = (x - 2)(x + 2)\), so the fraction is ready to split.

    \[\begin{aligned} &\frac{2x^3 + x^2 - 4x + 5}{x^2 - 4} \\ &\quad = \cq{2x + 1} + \frac{4x + 9}{(x - 2)(x + 2)} \end{aligned}\]

Practice

Prepare \(\dfrac{x^3 - 3x + 5}{x^2 - 3x + 2}\) for partial fractions.

Drag the handle to the right. The curve becomes close to a straight line, which long division finds. Then factor the denominator.

Solution

  1. The numerator has degree 3 and the denominator has degree 2, so divide. The quotient \(x + 3\) is the faint line, and the remainder is \(4x - 1\).

    \[\begin{aligned} &x^3 - 3x + 5 \\ &\quad = (x + 3)(x^2 - 3x + 2) \\ &\qquad + (4x - 1) \end{aligned}\]

  2. Factor the denominator. The roots 1 and 2 make it 0, because \(1 - 3 + 2 = 0\) and \(4 - 6 + 2 = 0\), and the dashed lines stand there.

    \[x^2 - 3x + 2 = (x - 1)(x - 2)\]

  3. The dotted curve is the remainder fraction. It is proper, and its denominator has two linear factors, so the fraction is prepared.

    \[\begin{aligned} &\frac{x^3 - 3x + 5}{x^2 - 3x + 2} \\ &\quad = \cq{x + 3} + \frac{4x - 1}{(x - 1)(x - 2)} \end{aligned}\]

Questions

The figure shows the curve of \(\dfrac{2x^2 - 7x + 9}{x^2 - 5x + 4}\). The dashed vertical lines mark where the denominator is 0.

  1. Which factorization of the denominator matches the dashed lines?

    • \((x - 1)(x - 4)\)
    • \((x + 1)(x + 4)\)
    • \((x - 1)(x + 4)\)
    Answer

    \((x - 1)(x - 4)\)

  2. Far to the left and right, the curve levels off near \(y = 2\). What is the quotient when you divide the numerator by the denominator?

    • \(2\)
    • \(0\)
    • \(x\)
    Answer

    \(2\)

  3. Divide \(x^3 + 2x^2 + 4\) by \(x^2 - 1\). What is the result?

    • \(x + 2 + \dfrac{x + 6}{x^2 - 1}\)
    • \(x + 2 + \dfrac{4}{x^2 - 1}\)
    • \(x + 2 + \dfrac{-x + 2}{x^2 - 1}\)
    • \(x + 2\)
    Answer

    \(x + 2 + \dfrac{x + 6}{x^2 - 1}\)

Summary

If the numerator has at least the degree of the denominator, you divide, and the curve follows the quotient line far from the dashed lines.

The remainder fraction is proper, and you factor its denominator into linear factors, whose roots are the dashed lines.

Do these two steps before you split a rational function into partial fractions.

This is one of four skills in topic 6.12 of Which & Why: AP Calculus.