Chapter 1 · Algebraic Proofs · Concept 3
1-6 Multiplication and Division of Rational Expressions
When a patient takes a medicine, the amount in their blood rises quickly and then falls as the body clears it. Doctors model that with fractions that have algebra on the top and the bottom — and to combine two treatments, they have to multiply and divide those fractions.
Guiding question
How do the ordinary rules of fractions let us combine and simplify algebraic fractions?
Learning outcome. Simplifies fractional (rational) expressions by doing multiplication and division.
A rational expression is just a fraction with algebra on the top and bottom, like (x + 1) / (x − 2). Everything you already do with 6/8 works here too.
Reducing
6/8 = 3/4
Divide top and bottom by 2. With algebra you divide by a common factor instead.
Multiplying
2/3 × 3/5 = 2/5
Cancel first, then multiply what is left. Never expand the brackets.
Dividing
2/3 ÷ 5/7 = 2/3 × 7/5
Turn the second one upside down and multiply. Exactly the same here.
Reducing
Dividing the top and the bottom by their common factor is called reducing. A rational expression that cannot be reduced any further is called irreducible.
(x − 1)(x − 3) / (x(x − 3)) → (x − 1) / x
The (x − 3) is on both rows, so it divides out. What is left cannot be reduced again.
When there are polynomials on the top or bottom, first factorise them to see whether anything can be reduced.
Here are the book's expressions, already factorised. Tap a block on the top that also appears on the bottom, and watch them go together. If you tap one that is not underneath as well, it will tell you why nothing happened.
The expression
(x²−4) / (3x²+7x+2)
Factorised, it looks like this. Tap a block on the top that also appears on the bottom.
A student cancels the x in this. Are they right?
(x + 4) / (x + 2)
1(b). Simplify (x2 − 4) / (3x2 + 7x + 2)
- 1
Do not cancel anything yet.
(x2 − 4) / (3x2 + 7x + 2)Nothing here is multiplying yet. Everything is added or subtracted, so nothing may go.
2. Calculate (x2 − 4x − 5)/(2x2 + 5x + 2) ÷ (x2 − x − 20)/(2x2 − 5x − 3)
- 1
Turn the dividing into multiplying.
× (2x2 − 5x − 3)/(x2 − x − 20)Flip the second fraction upside down. Exactly what you do with ordinary fractions.
Work with a partner. Do question 2 together, then talk about the answer. Remember: turn a division into a multiplication before you cancel anything.
1. Simplify.
2. Calculate.
1. Simplify.
2. Calculate.
If x = √123, find the value of
x³ + 8x² − x − 6×x² − 6x + 9x² − 2x + 4÷2x − 6x⁵ − 1
You may only cancel a FACTOR.
In (x + 4)/(x + 2) the x is being added, not multiplying. Crossing it out turns 5/4 into 2 at x = 6. If you are unsure, put a number in and check.
Factorise before you cancel anything.
Nothing can be cancelled from x² − 4 over 3x² + 7x + 2 until both are written as brackets multiplied together. Factorising is always the first move.
Flip before you cancel, not after.
With a division, turn the second fraction upside down first. Cancelling across a ÷ sign cancels the wrong things.
Do not expand the brackets at the end.
The book says so directly. (x + 1)(x − 3)/((x + 2)(x + 4)) is the finished answer — multiplying it out hides the factors and earns nothing.
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