Algebra, Year 11: factorising and algebraic fractions

A difference of two squares

提示The name uses the word for the result of a subtraction.

为什么Examples are x² − 9 and 100 − y². Every expression of this type factorises into two brackets that differ only in the sign: a² − b² = (a + b)(a − b).

(x + 7)(x − 7)

提示Both terms are squares, and one is taken from the other.

为什么49 = 7², so x² − 49 = (x + 7)(x − 7). Check by expanding: x² − 7x + 7x − 49, and the middle terms cancel.

The two middle terms are −5x and +5x, which add to zero

提示Multiply out all four products and look at the pair in the centre.

为什么x × x = x², x × (−5) = −5x, 5 × x = +5x and 5 × (−5) = −25. The −5x and +5x cancel, leaving x² − 25. That cancelling is exactly why a² − b² factorises so neatly.

200

提示Factorise first: one bracket adds the two numbers, the other subtracts them.

为什么51² − 49² = (51 + 49)(51 − 49) = 100 × 2 = 200. Factorising turns two awkward squarings into one easy multiplication.

x³ + 6x² + 11x + 6

提示Multiply two of the brackets together first, then multiply the result by the third.

为什么(x + 1)(x + 2) = x² + 3x + 2. Then (x² + 3x + 2)(x + 3) = x³ + 3x² + 3x² + 9x + 2x + 6 = x³ + 6x² + 11x + 6. Check with x = 1: 2 × 3 × 4 = 24 and 1 + 6 + 11 + 6 = 24.

x² − 3x − 4; 2; 19

提示Deal with two brackets at a time.

为什么(x + 1)(x − 4) = x² − 4x + x − 4 = x² − 3x − 4. Then each of its three terms is multiplied by x and by 5: x³ + 5x² − 3x² − 15x − 4x − 20, which collects to x³ + 2x² − 19x − 20.

30

提示Only one product of three numbers contains no x at all.

为什么The constant term comes from multiplying the three number terms: (−2) × (−5) × 3 = 30. Two negatives multiply to a positive. The full expansion is x³ − 4x² − 11x + 30.

8, because each term is one choice from each bracket: 2 × 2 × 2

提示Think of the product rule for counting.

为什么Every term in the expansion is made by picking one letter from each bracket, for example a × d × e. With two choices in each of three brackets there are 2 × 2 × 2 = 8 products. Counting them is a good check that none has been missed.

x³ + 6x² + 12x + 8

提示Write the bracket three times and multiply them out two at a time.

为什么(x + 2)² = x² + 4x + 4. Then (x² + 4x + 4)(x + 2) = x³ + 2x² + 4x² + 8x + 4x + 8 = x³ + 6x² + 12x + 8. Testing x = 1 shows Mia is wrong: 3³ = 27, but 1 + 8 = 9.

(2x + 1)(x + 3)

提示The x terms in the brackets must multiply to 2x², and the numbers to 3; test which arrangement gives 7x.

为什么The brackets start (2x …)(x …), and the numbers are 1 and 3. Trying (2x + 1)(x + 3): the middle terms are 6x + x = 7x, which is right. The other order, (2x + 3)(x + 1), gives 5x.

ac

提示It is not simply the constant term this time; the number in front of x² matters too.

为什么For 2x² + 7x + 3, a × c = 6, and the numbers 6 and 1 add to 7. Then 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3). When a = 1 this becomes the familiar 'multiply to c' rule.

(2x + 5)(2x − 5)

提示Each of the two terms is a perfect square.

为什么4x² = (2x)² and 25 = 5², so this is a difference of two squares: (2x)² − 5² = (2x + 5)(2x − 5). The 4 has to be square-rooted along with the x².

2(x + 2)(x + 3)

提示Look for a number that divides into every term before you start on brackets.

为什么Take out the common factor first: 2(x² + 5x + 6). The quadratic inside is then a simple one: 2(x + 2)(x + 3). 'Fully' is the signal that there is more than one step.

4

提示The two number terms in the brackets must multiply to give the constant term.

为什么2 × (−k) must equal −8, so k = 4. Check the middle term: 3x × (−4) + 2 × x = −12x + 2x = −10x, as required.

x/(x − 3)

提示Factorise the top and the bottom, then look for a bracket they share.

为什么The top is x(x + 3) and the bottom is (x + 3)(x − 3). The common factor (x + 3) cancels, leaving x/(x − 3).

(3x + 1)/(x(x + 1)), which is the same as (3x + 1)/(x² + x)

提示The common denominator is the two denominators multiplied together.

为什么1/x = (x + 1)/(x(x + 1)) and 2/(x + 1) = 2x/(x(x + 1)). Adding the numerators: x + 1 + 2x = 3x + 1. It is the same method as adding 1/3 + 2/5.

2/x

提示Dividing by a fraction is the same as multiplying by that fraction turned upside down.

为什么(x/4) × (8/x²) = 8x/(4x²). Cancelling the common factor 4x from top and bottom leaves 2/x.

Only factors of the whole top and the whole bottom can be cancelled, and x is a term that is added, not a factor

提示Try x = 2 in the original and compare with Tom's answer.

为什么With x = 2 the fraction is 8/4 = 2, not 3, so the 'simplified' form is not equal to the original. Cancelling is dividing top and bottom by the same thing, and (x + 6) ÷ x is not 6. This fraction cannot be simplified at all.

factorise

提示You need both the top and the bottom written as products of brackets.

为什么(x² − 4)/(x² + 5x + 6) = (x + 2)(x − 2)/((x + 2)(x + 3)) = (x − 2)/(x + 3). Only once both are written as products can a common factor be seen and cancelled.