ABSTRACT
Usually when we begin to study algebra at school we first meet and work with expressions containing a letter, like x – 1 or 4x 2 – 9x + 2, before we learn that such expressions are called ‘polynomials’ † . We commonly think of x as just standing in place of a number; so that in 4x 2–9x + 2 for instance we could replace x by 5 and find that the value of the polynomial is then 4 * 52 – 9 * 5 + 2= 57. We could actually write out ‘four times the square of a given number minus nine times the given number plus two’ but that is more difficult to remember, and work with, than the shorthand 4x 2–9x + 2. Medieval mathematicians often did write out expressions in words; only they wrote in Latin. Similarly the value of https://www.w3.org/1998/Math/MathML"> x 2 + 2 3 x − 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003420194/47ed985f-4dde-48d0-bcde-32592266d8c3/content/ieq0001.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> when x is replaced by 5 is https://www.w3.org/1998/Math/MathML"> 5 2 + 2 3 * 5 − 1 = 27 1 3 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003420194/47ed985f-4dde-48d0-bcde-32592266d8c3/content/ieq0002.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . Note that https://www.w3.org/1998/Math/MathML"> ( 4 * 5 2 − 9 * 5 + 2 ) + ( 5 2 + 2 3 * 5 − 1 ) = 5 * 5 2 − 8 1 3 * 5 + 1 = 84 1 3 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003420194/47ed985f-4dde-48d0-bcde-32592266d8c3/content/eqn0001.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> so that the value of https://www.w3.org/1998/Math/MathML"> 5 x 2 − 8 1 3 x + 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003420194/47ed985f-4dde-48d0-bcde-32592266d8c3/content/ieq0003.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> when x = 5 is https://www.w3.org/1998/Math/MathML"> 84 1 3 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003420194/47ed985f-4dde-48d0-bcde-32592266d8c3/content/ieq0004.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . We could express this by putting https://www.w3.org/1998/Math/MathML"> ( 5 x 2 − 8 1 3 x + 1 ) x = 5 = ( 4 x 2 − 9 x + 2 ) x = 5 + ( x 2 + 2 3 x − 1 ) x = 5 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003420194/47ed985f-4dde-48d0-bcde-32592266d8c3/content/eqn0002.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and likewise if x were replaced by 3 say we would have https://www.w3.org/1998/Math/MathML"> ( 5 x 2 − 8 1 3 x + 1 ) x = 3 = ( 4 x 2 − 9 x + 2 ) x = 3 + ( x 2 + 2 3 x − 1 ) x = 3 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003420194/47ed985f-4dde-48d0-bcde-32592266d8c3/content/eqn0003.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> An analogous equation would also hold if, in each of https://www.w3.org/1998/Math/MathML"> 5 x 2 − 8 1 3 x + 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003420194/47ed985f-4dde-48d0-bcde-32592266d8c3/content/ieq0005.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , 4x 2-9x+2 and https://www.w3.org/1998/Math/MathML"> x 2 + 2 3 x − 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003420194/47ed985f-4dde-48d0-bcde-32592266d8c3/content/ieq0006.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , we replaced x by any other number instead of 3 or 5. Of course it gets very tedious to write down separate and closely similar equations such as https://www.w3.org/1998/Math/MathML"> ( 5 x 2 − 8 1 3 x + 1 ) x = 7 = ( 4 x 2 − 9 x + 2 ) x = 7 + ( x 2 + 2 3 x − 1 ) x = 7 ( 5 x 2 − 8 1 3 x + 1 ) x = − 2 = ( 4 x 2 − 9 x + 2 ) x = − 2 + ( x 2 + 2 3 x − 1 ) x = − 2 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003420194/47ed985f-4dde-48d0-bcde-32592266d8c3/content/eqn0004.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> We could conveniently summarise all such equations by writing https://www.w3.org/1998/Math/MathML"> 5 x 2 − 8 1 3 x + 1 = ( 4 x 2 − 9 x + 2 ) + ( x 2 + 2 3 x − 1 ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003420194/47ed985f-4dde-48d0-bcde-32592266d8c3/content/eqn0005.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> meaning that when we replace x throughout (1.1) by any fixed number we obtain a true statement about numbers. In a similar way we could write https://www.w3.org/1998/Math/MathML"> 3 x 2 − 9 2 3 x + 3 = ( 4 x 2 − 9 x + 2 ) − ( x 2 + 2 3 x − 1 ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003420194/47ed985f-4dde-48d0-bcde-32592266d8c3/content/eqn0006.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> meaning again that when we replace x everywhere in (1.2) by a fixed number we obtain a true statement about numbers. For example, if we choose https://www.w3.org/1998/Math/MathML"> x = − 11 , ( 3 x 2 − 9 2 3 x + 3 ) x = − 11 = 472 1 3 , ( 4 x 2 − 9 x + 2 ) x = − 11 = 585 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003420194/47ed985f-4dde-48d0-bcde-32592266d8c3/content/ieq0007.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , https://www.w3.org/1998/Math/MathML"> ( x 2 + 2 3 x − 1 ) x = − 11 = 112 2 3 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003420194/47ed985f-4dde-48d0-bcde-32592266d8c3/content/ieq0008.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and indeed https://www.w3.org/1998/Math/MathML"> 472 1 3 = 585 − 112 2 3 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003420194/47ed985f-4dde-48d0-bcde-32592266d8c3/content/ieq0009.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> in agreement with (1.2).
