33. x^2 + xy + y^ and x — y. 34. x2 - ax - a2 and x^2 - ax + a^2. 36. 1 x^2 - x - and x^2 + x - . 36. ax + x^2 + 1a^2 and a^2 + 3 x^2 - 3 ax.
Note. Examples involving literal, fractional, and negative exponents will be found in the chapter on the Theory of Indices.
51. Products Written by Inspection. Although the result of multiplying together two binomial factors, such as x+8 and x — 7, can always be obtained by the methods already explained, it is of the utmost importance that the student shoud learn to write the product rapidly by inspection.
This is done by observing in what way the coefficients of the terms in the product arise, and noticing that they result from the combination of the numerical coefficients in the two binomials which are multiplied together ; thus
(x + 8) (x + 7)= x^2 + 8x + 7x + = x^2 + 15x- 56. (x- 8) (x - 7)= x^- 8x - 7x + = x^2-15x + 56. (x + 8) (x - 7)= x^2 + 8x - 7x - 56 = x^2 + x — 56. (x - 8) (x + 7)= x^2 - 8x + 7x - = x^2 — x — .
In each of these results we notice that : 1. The product consists of three terms. 2. The first term is the product of the first terms of the two binomial expressions. 3. The third term is the product of the second terms of the two binomial expressions. 4. The middle term has for its coefficient the sum of the numerical quantities (taken with their proper signs) in the second terms of the two binomial expressions.
