Partial Fractions and Factoring on the TI-89

Partial fractions on the TI-89 take one command, but it is not the one most people look for. This guide shows how to factor a polynomial on the TI-89, how to split a rational function into partial fractions, and how to move between the forms your homework asks for.

There is no simplify button

The TI-89 Titanium simplifies every result automatically when you press ENTER. What it cannot guess is which form you want: factored, expanded, over a common denominator or split into partial fractions. Each of those has its own command in the F2 Algebra menu on the HOME screen:

Rewriting commands in the F2 Algebra menu
Command	Gives	Keys
factor(	Product of factors	`F22`
expand(	Sum of terms, partial fractions	`F23`
comDenom(	One fraction over a common denominator	`F26`
propFrac(	Whole part plus proper fraction	`F27`
cFactor(	Factors including complex ones	`F2►►2`

How to factor a polynomial

Press F2 and 2 to paste factor(, type the polynomial and close the parenthesis. factor( looks for rational factors, so it works over the real numbers you know from algebra class.

F22→ factor(

factor( examples
Input	Result
factor(x^2-1)	(x−1)·(x+1)
factor(x^4-1)	(x−1)·(x+1)·(x²+1)

The second row shows the limit of real factoring: x² + 1 has no real roots, so it stays as it is. If an expression has more than one letter, give the variable as a second argument, for example factor(expression, x), so the CAS knows which letter to factor in.

factor( also works on a plain integer and returns its prime factorization, which is handy for simplifying roots and fractions by hand.

Complex factors with cFactor(

cFactor( goes one step further and factors over the complex numbers. It is in the Complex submenu of F2.

F2►►2→ cFactor(

cFactor( example
Input	Result
cFactor(x^2+1)	(x+ⅈ)·(x−ⅈ)

Expanding

expand( is the opposite of factor(: it multiplies everything out into a sum of terms. Use it to check a factorization or to compare your hand calculation with the CAS.

F23→ expand(

expand( example
Input	Result
expand((x+1)^2)	x²+2·x+1

Partial fractions on the TI-89 with expand(

Here is the part most guides hide: partial fraction decomposition is expand( with a rational expression and the variable. Type the fraction with parentheses around numerator and denominator, then a comma and x:

expand((numerator)/(denominator),x)

The CAS factors the denominator, splits the expression into a sum of simpler fractions and, if the degree of the numerator is not smaller than the degree of the denominator, puts the polynomial part in front. The variable argument tells the CAS which letter to decompose in, so always give it.

A few tips that save time:

Parentheses. (x+1)/(x^2-1) is not the same as x+1/x^2-1. Wrap both parts.
Check by recombining. comDenom( on the result should give back your original fraction, possibly in a different but equivalent form.
Irreducible quadratics. A factor like x² + 1 in the denominator stays a quadratic term with a linear numerator, as you would write it by hand.
For integrals. You rarely need the decomposition to integrate on the TI-89, because ∫( handles rational functions directly. Use expand( when the problem asks for the partial fractions themselves.

Common denominators and proper fractions

comDenom( does the reverse of partial fractions: it combines a sum into a single fraction. propFrac( splits a fraction into a whole part and a proper fraction, for numbers and for rational expressions.

F26→ comDenom(

F27→ propFrac(

comDenom( and propFrac( examples
Input	Result
comDenom(2+1/x)	(2·x+1)/x
propFrac(7/3)	2+1/3

To take a fraction apart, getNum( and getDenom( return the numerator and the denominator.

Numerator and denominator
Input	Result
getNum(3/(x+2))	3
getDenom(3/(x+2))	x+2

Polynomial division and completing the square

Calc89 adds a few commands for these steps that you would otherwise do by hand on the handheld: polyQuotient( and polyRemainder( for long division, and completeSquare( for vertex form.

Polynomial division and completing the square in Calc89
Input	Result
polyQuotient(x^3-1,x+1,x)	x²−x+1
polyRemainder(x^3-1,x+1,x)	−2
completeSquare(x^2+3x,x)	(x+3/2)²−9/4

When factor( returns a number

The most common forum question about factoring on the TI-89: factor( suddenly returns a number instead of factors. The reason is almost always a stored variable. If x holds a value from an earlier problem, the CAS substitutes it before factoring. Delete it with DelVar x, or run NewProb from the Clean Up menu (2nd F1) before each new problem. The same applies to expand( and solve(.

Two more things to check: write x*y, not xy, which is a single variable; and in APPROXIMATE mode, coefficients come out as decimals, so keep Exact/Approx on AUTO for algebra.

In Calc89

Calc89 is intended for use with TI-89 Titanium, and the F2 Algebra menu uses the same items, so F2 2 is factor( and F2 3 is expand( on your phone too. Every result in the tables above comes from the same CAS that runs in the app.

Rational functions are tedious to type. Point the camera scanner at the problem, printed or handwritten, and it lands in the edit line with the parentheses in place. The AI keystroke tutor then shows which command gives the form you need and the keys for it, step by step.

Frequently asked questions

How do you do partial fractions on the TI-89?

Use expand( with the variable as second argument, for example expand(rational expression, x). The CAS splits the fraction into a sum of simpler fractions.

Why does factor( give me a number instead of factors?

The variable has a stored value. Delete it with DelVar or run NewProb, then factor again.

Is there a simplify command on the TI-89?

No. The TI-89 simplifies every result automatically when you press ENTER. Use factor(, expand(, comDenom( or propFrac( to choose the form.

Partial Fractions, Factoring and Simplifying on the TI-89