Adding and subtracting algebraic fractions

common denominator. Consider the addition . The lowest common denominator (LCD) is not necessarily found by multiplying the denominators. Rather, it is found by factorising the individual denominators into primes, with each different factor then taken once and multiplied together.

For example, =

=

=

=

The same method is used to add or subtract algebraic fractions.

NOTE: If one of the denominators is a perfect square, both factors must be included in the LCD.

Example 1

Adding and subtracting algebraic fractions

---To add or subtract algebraic fractions with binomial or trinomial denominators:

factorise the denominator in each fraction if possible

form the LCD by taking each different factor in the individual denominators once and finding their product

divide the denominators into the LCD then multiply by the numerators

add or subtract the numerators

check whether the resulting fraction can be simplified by factorising and cancelling.

---Example 2

1 Simplify each of the following.

a b c d

2 Simplify:

5 Simplify each of the following.

a b

6 Factorise the denominator in each fraction, then express the fractions with a common denominator and simplify.

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7 Factorise the denominator in each fraction, then express the fractions with a common denominator and simplify.

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Introduction

What are ‘taxicab numbers’? How did they come be studied? Who were the mathematicians involved? We can only relate part of the story here.

It began during the first world war when the British mathematician GH Hardy (1877–1947) went to visit his protégé and colleague Srinivasa Ramanujan (1877–1920) who lay dying in hospital in London.

Hardy had gone out to the hospital in Putney by taxi.

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OCUS ON WORKING MATHEMATICALLY He was a shy and self conscious man in situations like this, not knowing how to open a

conversation easily, despite the fact that he had worked with Ramanujan for years. CP Snow in the foreword to Hardy’s book (A Mathematicians Apology, Cambridge University Press, 2000, page 37) records the conversation. Without a greeting, and certainly as his first remark, Hardy blurted out:

‘I thought the number of my taxicab was 1729. It seemed to me a rather dull number.’

To which Ramanujan replied, ‘No Hardy! No Hardy! It is a very interesting number.

It is the smallest number expressible as the sum of two cubes in two different ways.’

The number 1729 can be written as 1³+ 12³ and also as 9³ + 10³ . Today such numbers have become known as ‘taxicab numbers’. Mathematicians define the smallest number expressible as the sum of two cubes in n different ways as the nth taxicab number, denoted by taxicab(n).

Thus taxicab(2) = 1729. Taxicab(3) was discovered in 1957 to be 87 539 319 and taxicab(4) in 1991 to be 6 963 472 309 248. Taxicab(5) was discovered in 1997. As you can imagine, computers played a major role in the discovery of taxicab numbers but little else is known about them.

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2 Let’s take a closer look at taxicab(2) = 1729. Using a calculator verify that 1729 can be written as the sum of the two cubes 1³+ 12³ and 9³+ 10³ .

3 In this chapter you have learnt to factorise algebraic expressions. By multiplying out, verify that the factors of the expression a³+ b³= (a + b)(a²− ab + b²).

4 Let a= 9 and b = 10. Use your calculator to verify that the right hand side is 1729. Repeat with a= 1 and b = 12.

5 Equations of the form c= a³+ b³ are called Diophantine equations (named after

Diophantus, around 250 AD) where a, b, and c are integers. Solve the Diophantine equation 28 = a³+ b³ for a and b. Why is 28 not a taxicab number?

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1 Two cubes have side lengths a and b (whole numbers with a< b). The sum of their volumes is equal to the sum of the lengths of their edges. Find a and b.

2 Given that a³+ b³= (a + b)(a²− ab + b²) deduce the factors of a³− b³ by setting −b for b.

3 Two cubes have side lengths a and b (whole numbers with a> b). The difference in their volumes is equal to the difference of the total lengths of their edges. Investigate whether integer solutions for a and b can be found.

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Reasoning means making logical statements in a sequence. To combine logical statements, we can use ‘linking words’ such as and, if, when, however, because or but to name a few. Write a short paragraph using linking words to communicate the meaning of taxicab numbers. If you have tried this you will immediately see the power of algebra to present meaning in symbolic form.

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There are two ways to describe mathematical thinking on which you should reflect. One is through a search for specific patterns which may suggest a general rule. This type of thinking is inductive. The second concerns the need for proof. In this case the result we suspect to be true is put to the test of deductive reasoning, that is a rigorous chain of argument that leads to an inevitable conclusion.

In their book An Introduction to the Theory of Numbers (Oxford University Press, 1954), GH Hardy and his colleague EM Wright proved a theorem to show that taxicab numbers, denoted by taxicab(n), exist for any value of n≥ 1. What type of mathematical reasoning do you think they used?

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Use each of the following in a simple sentence:

1 Binomial factor

2 Difference of two squares 3 The sum of two cubes

4 The difference between inductive and deductive reasoning

5 The Macquarie Learners Dictionary entry for factor:

factor noun 1. one of the things that brings about a result: Hard work was a factor in her success.

2. Specialised one of two or more numbers which, when multiplied together, give the product: Factors of 18 are 3 and 6.

Note the special mathematical meaning of factor and that it applies in algebra as well as in arithmetic.