61. Geometric Theorems on Factoring. Addition of rectangles. As in Chapter IX, we can express multiplication of polynomials as a combination of rectangles. Draw the rectangle whose base is be and whose height is h units. Thus: Is the area of the rectangle h (b + e) equal to the sum of the area of rectangles hb and he? Write the algebraic equation for this. We assume here and in what follows that the letters stand for positive numbers. Do similarly with m (h + k + 1). We can state this as a theorem. Theorem 1. The rectangle of two given lines is equal to the sum of the rectangles of one of them and the several segments into which the other is divided. Exercises. Draw the following and write the algebraic expression for the equality of areas. Definition. When an expression like c2+2c+3 cd is rewritten in the form of a product, c (c + 2 + 3 d), this operation is called factoring. The above exercises are simple exercises in factoring, illustrated geometrically. Theorem 2. The square on the sum of two lines equals the sum of the squares on those lines plus twice their rectangle. We have given the square on the sum of the two lines a and b. Call it OKTM. To show that square OKTM is equal to two squares, one of which has a side a, the other has a side b, and two rectangles of the dimensions a by b. In other words we are to show that (a + b)2 = a2 + 2 ab + b2. Name the parts of which the square OKTM is composed and give the dimensions of each. Make the algebraic multiplication of (a+b) (a+b). Show the correspondence in the figure of each partial product. Exercises. Make a geometric representation and algebraic expression of the following: Give the algebraic expression of the following without drawing: 10. m2 + 4 m + 4. 11. 96 h + h2. 12. 9 r2 + 12 rs + 4 r2. On the same principle as above give the geometric picture of the following rectangles, give the algebraic solution, and point out the geometric representation of each step in the solution. Subtraction of rectangles. Draw the rectangle h (b − e). Do this by drawing length b, then subtract length e, e being shorter than b. Draw height h and complete the rectangle. Thus: The rectangle MNQP is the rectangle h (be). Rectangle ONQS is rectangle hb; OMPS is rectangle he; rectangle OMPS minus rectangle ONQS is rectangle MNQP. Do the same with m (h k +1); that is, show that this is equivalent to rectangle mh, minus rectangle mk, plus rectangle ml. Exercises. Resolve the following into simpler rectangles: 1. m (a - b). 2. 5 (2s+rt). 3. c (de 3 c). Theorem 3. The square on the difference between two lines equals the sum of the squares on the two lines minus twice the rectangle of the lines. To draw the square whose side is (a - b), draw length a; from it subtract length b. The remainder ab is the base. altitude a In like manner draw the square. (See figure on next page.) b, so as to form a The resulting figure PQRS is the square (a - b) (a - b). OMRT is the square on a. Its area is a2. OMQH is the rectangle ab. OKST is the rectangle ab. Rectangles OMQH and OKST are both subtracted from square OMRT. But notice that after subtracting rectangle OMQH it will be necessary to add square OKPH before we can subtract rectangle OKST, so that rectangle OMRT minus rectangle Exercises. Draw and give algebraic expressions for the following: Write the algebraic expressions for the following without drawing; in other words, factor the given expressions: 7. d2 - 2 de + e2. 8. 96r+ r2. 9. 168s s2. 10. 25 t2 - 30 t + 9. 11. 9 m2n2 - 24 mn + 16. |