23. The signs of aggregation are: the parenthesis, (); the bracket, [ ]; the brace, {}; and the vinculum, They are used, as in arithmetic, to indicate that the expressions included are to be treated as a whole. Each of the forms 10 × (4 + 1), 10 × [4 + 1], 10 x 4 + 1 indicates that 10 is to be multiplied by 4+1 or by 5. (a - b) is sometimes read "quantity a — b.” EXERCISE 7 If a=2, b = 4, c = 1, d= 0, x=9, find the numerical value of: 18. In Ex. 14 what is the coefficient of the √b? of (a+b)√õ? ALGEBRAIC EXPRESSIONS AND NUMERICAL SUBSTITUTIONS 24. An algebraic expression is a collection of algebraic symbols representing some number; e.g. 6 ab-7√ac2+9. 25. A monomial or term is an expression whose parts are not separated by a sign + or - ; as 3 ax2, -9√x, 3 ab 5 c2 a(b + c + d) is a monomial, since the parts are a and (b + c + d). 26. A polynomial is an expression containing more than one term. + √z − 3 a3b; and aa + ba + ca + d1 are polynomials. y 27. A binomial is a polynomial of two terms. a2 + b2, and § - Va are binomials. 28. A trinomial is a polynomial of three terms. a+b+c, a + 9b + √3 are trinomials. 29. In a polynomial each term is treated as if it were contained in a parenthesis, i.e. each term has to be computed before the different terms are added and subtracted. Otherwise all operations of addition, subtraction, multiplication, and division are to be performed in the order in which they are written from left to right. E.g. 3+4.5 means 3 + 20 or 23. Ex. 1. Find the value of 4·23 + 5 · 3o — 9√36 Ex. 2. If a = 5, b = 3, c = 2, d=0, find the numerical value of 6 ab2-9 ab2c+ab-19 a2bcd. 1. State what kind of expressions are Exs. 18-27 of this exercise. = If a=5, b 2, c=1, d=0, x= = 1, and y=1, find the numerical value of: 28. 7 cd-x(3 abx + c2)− cx(d+b) — 12 c2x. 34. (a+b) (c+d) − (a + c) (b + d) + (a + d) (b − c). Express in algebraic symbols: 35. Six times a plus 3 times b. 36. Six times the square of a minus five times the cube of b. 37. Eight a cube minus six x cube plus y square. 38. Six m cube plus four times the quantity a minus b. 39. The quantity a plus b multiplied by the quantity a2 minus b2. 40. Twice a diminished by 5 times the square root of the quantity a minus b square. 41. Read the expressions of Exs. 2-6 of the exercise. 30. The representation of numbers by letters makes it possible to state very briefly and accurately some of the principles of arithmetic, geometry, physics, and other sciences. Ex. If the three sides of a triangle contain respectively a, b, and c feet (or other units of length), and the area of the triangle is S square feet (or squares of other units selected), then S = {√(a + b + c) (a + b − c) ( a − b + c) (b − a + c). E.g. the three sides of a triangle are respectively 13, 14, and 15 feet, then a = 14, and c = = 15; therefore 13, b = S=1√(13+14+15)(13+14−15) (13−14+15) (14−13+15) S = {√(a + b + c) (a + b − c) ( a − b + c) (b − a + c), find the area of a triangle whose sides are respectively (a) 5, 12, and 13 feet. (b) 3, 4, and 5 inches. (c) 4, 13, and 15 meters. (d) 9, 10, and 17 yards. = = 3.1416 R2 square units (square 2. If the radius of a circle is R units of length (inches, meters, etc.), the area S inches, square meters, etc.). radius is Find the area of a circle whose (a) 1000 meters. (b) 3 inches. (c) 240,000 miles. 3. If i represents the simple interest of p dollars at r% in n years, then i=p•n•r%, or Find by means of this formula : (a) The interest on $730 for 4 years at 21%. 4. If I represents the compound interest of p dollars at r% for n years (compounded yearly), then I=p(1+ n p(1. 100) * - p. Find the compound interest of: (a) $400 for 3 years at 10%. 5. If the diameter of a sphere equals d units of length, the surface S3.1416 d (square units). (The number 3.1416 is frequently denoted by the Greek letter . This number cannot be expressed exactly, and the value given above is only an approximation.) Find the surface of a sphere whose diameter equals : (a) 8000 miles. (b) 2 inches. (c) 12 feet. 6. If the diameter of a sphere equals d feet, then the volume Find the volume of a sphere whose diameter equals: (a) 10 feet. = 7. A body falling from a state of rest, passes in t seconds over a space S gt. The value of g for New York is 32,16 feet, or 980 cm. (This formula does not take into account the resistance of the atmosphere.) (a) How far does a body fall from a state of rest in 3 seconds? (b) A stone dropped from the top of a tree reached the ground in 2 seconds. Find the height of the tree. |