Chapter 1 · Algebraic Proofs · Concept 3
1-7 Addition and Subtraction of Rational Expressions
In lesson 1-6 you multiplied and divided these fractions, and cancelling did most of the work. Adding is different: nothing can be cancelled until the two bottoms are the same.
Guiding question
How do you add two algebraic fractions whose bottoms are not the same?
Learning outcome. Simplifies fractional (rational) expressions by doing addition and subtraction.
1Factorise the bottoms
You cannot see what they share until they are in brackets.
2Build the common denominator
Every bracket that appears anywhere, each used once.
3Work out the TOP only
The bottom is already decided. Leave it alone.
4Factorise the top and reduce
Very often something cancels. The book’s examples nearly always do.
In that order, every time. Most lost marks in this lesson come from doing move 3 before move 2 — adding the tops while the bottoms are still different.
Adding fractions with different bottoms
Just as with ordinary fractions, multiply the top and bottom by the same thing to make the bottoms match, and only then add.
1/(x+1) + 1/(x−2) = (x−2)/((x+1)(x−2)) + (x+1)/((x−2)(x+1))
= ((x−2) + (x+1)) / ((x+1)(x−2)) = (2x−1) / ((x+1)(x−2))
When the bottoms are polynomials, factorise them first — that is the only way to see what they already share.
2/(x²−x−2) + 3/(x²−5x+6) = 2/((x+1)(x−2)) + 3/((x−2)(x−3))
= (2(x−3) + 3(x+1)) / ((x+1)(x−2)(x−3)) = (5x−3) / ((x+1)(x−2)(x−3))
The (x−2) is in both bottoms, so it is written once, not twice.
Once they are over one bottom, work out only the top. Then check whether the top factorises and something reduces.
A student adds the tops and the bottoms. Are they right?
1/(x + 1) + 1/(x − 2) = 2 / (2x − 1) ?
Multiplying the two bottoms together always works, but it makes them bigger than they need to be — and then you have to cancel it back at the end. Here is the difference.
The two bottoms are x² − x − 2 and x² − 5x + 6.
Simplify (x + 4)/(x2 − 2x) − 3/(x2 − 3x + 2)
- 1
Move 1 — factorise both bottoms.
= (x+4)/(x(x−2)) − 3/((x−1)(x−2))x² − 2x = x(x−2), and x² − 3x + 2 = (x−1)(x−2). Now you can see they share (x−2).
Work with a partner. Do these together, then talk about the answers. Say the four moves out loud as you go.
Simplify.
A team is watching a two-stage water filter over x hours. Stage 1 pulls pollutants out at a rate of (x³ + 8x)/(x² − x − 2). Stage 2 lets a little flow back, at 8/(x − 2). Subtract the Stage 2 rate from the Stage 1 rate to find how well the whole thing cleans.
Never add the bottoms.
1/(x+1) + 1/(x−2) is not 2/(2x−1). You knew at primary school that 1/2 + 1/3 is not 2/5, and nothing about algebra changes it.
Factorise the bottoms before choosing one.
x² + 5x − 6 already contains (x − 1). Multiplying by another (x − 1) you did not need makes the top twice as long and gives you more chances to slip.
Multiply each top by what ITS bottom was missing.
Not by the whole common denominator. If the bottom went from x(x−2) to x(x−1)(x−2), that top is multiplied by (x−1) only.
Always check the top at the end.
In nearly every question on page 18 the top factorises and something cancels. If yours does not, look again — you have probably made an arithmetic slip.
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