Fundamentals in Algebra - Basic Rules

Fundamentals in Algebra DAY 2 21 FUNDAMENTALS IN ALGEBRA BASIC RULES OF ALGEBRA Let a, b, and c be real numbers, variables or algebraic expressions. → Closure property Addition a + b Commutative property Multiplication ab : Addition a + b = b + a Associative property Multiplication ab = ba Addition (a + b) + c = a + (b + c) Identity property Multiplication (ab) * c = a(bc) Addition a + 0 = 0 + a = a Inverse property Multiplication a * 1 = 1a = a Addition a + (- a) = 0 = (- a) + a Multiplication a(1/a) = 1 = (1/a) * a with a ne0 Distributive property : Left a(b + c) = ab + ac : Right (a + b) * c = ac + bc In item 4, 0 is called an additive identity while 1 is called a multiplicative identity. In item 5, a is known as additive inverse of a while 1/a is the multiplicative inverse of a. PROPERTIES OF EQUALITY Let a, b and c be real numbers, variables of algebraic expression ① Reflexive property a = a Symmetric property : If a = b then b = a Transitive property : If a = b and b = c then a = c Substitution property if a = b then a can be replaced by b in any expression involving a Addition/Subtraction : If a = b then a + c = b + c Multiplication/Division : If a = b then a - c = b - c If a = b then ac = bc : If a = b then a/c = b/c with c ne0 MODERN ALGEBRA 1. Basic Theory of heal Numbers 2. Equation Functions & traphs Exponent & Exponential Fractions 24 1001 Solved Problems in Engineering Mathematics by Tiong & Rojas Sum of two cubes Square of a trinomial x ^ 3 + y ^ 3 = (x + y)(x ^ 2 - xy + y ^ 2) (x + y + x) ^ 2 = x ^ 2 + y ^ 2 + z ^ 2 + 2xy + 2xz 2yz PROPERTIES OF PROPORTION a) If a/y = x/d then a / x = y / d b) a/b = c/d then a/c = b/d c) a/b = c/d then b/a = d/c d) If a/b = c/d then (a - b)/b = (c - d)/d e) If a/b = d/d then (a + b)/b = (c + d)/d If a/b = c/d then (a + b)/(a - b) = (c + d)/(c - d) Note in item (a), quantities a & d are called extremes and x & y are called means. If x = y this is known as the mean proportional. In the ratio x/y, the first term x is antecedent while the second term y is called the consequent. yx = y the means are known as mean proportional extremes a / x = y / d ↑↑ means antecedent a / x = a/x consequent THE REMAINDER THEOREM & FACTOR THEOREM If a polynomial in an unknown quantity x is divided by a first degree expression in the same variable, (x-k), where k may be any real or complex number, the remainder to be expected will be equal to the sum obtained when the numerical value of k is substituted for x in the polynomial. Thus, Remainder = f(x) x -> k Fundamentals in Algebra 23 PROPERTIES OF RADICALS In the expression, root(a, n) , n is called the index, a the radicand while the symbol is called the radical or radical symbol. Properties: root(a ^ m, n) = (root(a, n)) ^ m root(a, n) * root(b, n) = root(ab, n) b ne0 (root(a, n))/(root(b, n)) = root(a/b, n) root(root(a, m), m) = root(a, m) root(a ^ n, n) = |a| root(a ^ n, n) = a (root(a, n)) = a Examples: root(8 ^ 2, 3) - (root(8, 3)) ^ 2 - (2) ^ 2 = 4 root(5, 4) * root(675, 3) = root(5(675), 3) = root(3375, 3) = 15 (root(50, 3))/(root(10, 3)) = root(50/10, 3) = root(5, 3) root(15, 4) = sqrt[6] 15 (root(2x, 5)) ^ 5 = 2x root((- 12) ^ 4, 4) = |- 12| = 12 For even ven number n = 1 root((- 15) ^ 3, 3) = - 15 For n = 1 odd number Surd is a radical expressing an irrational number. The surd is described after the index of the radical. For example, sqrt(3) is a quadratic surd, root(3, 3) is a cubic surd, root(3, 4) is a quartic surd and so on. Pure surd, sometimes called as entire surd contains no rational number and all its terms are surds. For example, sqrt(3) + sqrt(2) . A mixed surd contains at least one rational number. 5sqrt(3) is a mixed surd because 5 is rational number while sqrt(3) is a surd. A binomial surd is an expression of two terms with at least one term is a surd. For example, 5 + sqrt(2) A trinomial surd is an expression of three terms with at least two of them are and cannot be expressed as a single surd, otherwise it will become a binomial surd. Example, 5 + sqrt(2) + sqrt(3) SPECIAL PRODUCTS Let x, y and z be real numbers, variables or algebraic expression. Sum and difference of same terms: (x + y)(x - y) = x ^ 2 - y ^ 2 Or Difference of two squares Square of a binomial Cube of a binomial Difference of two cubes (x + y) ^ 2 = x ^ 2 + 2xy + y_{2} ^ 2 (x-y) 3 ^ 2 = x ^ 2 - 2xy + y ^ 2 (x+y) 3 ^ 3 = x_{3} ^ 3 + 3x ^ 2 * y + 3x * y_{2} ^ 2 + y_{3} ^ 3 (x - y) ^ 3 = x ^ 3 - 3x ^ 2 * y + 3x * y ^ 2 - y ^ 3 x ^ 3 - y ^ 3 = (x - y)(x ^ 2 + xy + y ^ 2) 22 1001 Solved Problems in Engineering Mathematics by Tiong & Rojas Cancellation property If a + c = b + c then a = b If acbc and c = 0 then a = b PROPERTIES OF ZERO Let a and b real numbers, variables or algebraic expression. a + 0 = a and a + 0 = a a o = 0 a = 0 0/a = 0 0 is undefined a/0 If ab = 0 then a = 0 or b = 0 This is known as Zero-Factor Property PROPERTIES OF EXPONENTS The exponential notation states that if a is a real number, variable, or algebraic expression and n is a positive number, then n factors Properties: a ^ m * a ^ n = a ^ (m + n) a^ n = aaa **** a (a ^ m)/(a ^ n) = a ^ (m - n) (a ^ m) ^ n = a ^ (mn) (ab) ^ m = a ^ m * b ^ m 6 (a/b) ^ m = (a ^ m)/(b ^ m) a ^ (m / n) = root(a ^ m, n) a ^ (- m) = 1/(a ^ m) a ne0 a deg = 1 Examples: 4 ^ 2 + 4 ^ 3 = 4 ^ (2 + 3) = 4 ^ 5 (x ^ 8)/(x ^ 3) = x ^ (8 - 3) = x ^ 5 (y ^ 6) ^ 2 = y ^ (6(2)) = y ^ 12 (2x) ^ 4 = 2 ^ 4 * x ^ 4 = 16x ^ 4 (2/x) ^ 4 = (2 ^ 4)/(x ^ 4) = 16/(x ^ 4) 4x ^ (5/3) = root((4x) ^ 5, 3) x ^ - 5 = 1/(y ^ 5) (x ^ 5 + 2) ^ 0 = 1 26 1001 Solved Problems in Engineering Mathematics by Tiong & Rojas Test 2 Time element: 3 hours & 45 minutes Problem 51: ECE Board April 1999 If 16 is 4 more than 4x, find 5x-1. A. 14- SOL zeta*ubK + y + ingx = 3 B. 3 I_{L} = 4x + 4 C. 12 I_{L} - 4 = 4x FX-1 D. 5 5(3)-1 = 14 x = 2 Problem 52: EE Board October 1992 Find the value of x in (x + 1)/3 + (2x)/4 = 47 - 2x A. 16.47- GL: [(x + 1)/3 + (2x)/4 = 47 - 2x] * 12 B. 12.87 C. 18.27 (x + 1)/3 + (2x)/4 = 47 - 2x D. 20.17 multiply all by 12 (124 + 12)/3 + (2dx)/4 = 564 - 24x 4x + 4 + 6x = 564 - 24x 4x + 6x + 24x =56 ne 34x =s60 x=| 6 - 47 x = 50/34 x = [140/18] * 2/2 19/x |= 18 16 x = 70/9 Problem 53 : EE Board October 1991 Find the value of x in the equations: (ultitating 122 6/x - 8(1/10 - 1/7) = 1 6/x - y/10 + 8/x =1-10[A/x + A/y] = A 2[(3A)/x - (4A)/y] = A (1/y = 1/10 - 1/x) - eq * 1 2[(3A)/x - (4A)/Y] = 4 1/x + 1/y = 1/10 14/x = 1 + q/10 (k/x - q/y = 1) - ey * 2 (10A)/x + (ln(A))/y = A ^ 7 * [1/(10A)] [(6A)/x - (4x)/y] = A \ 1 y A. 50/9 B. 80/9 C. 70/9- D. 60/9 Problem 54: EE Board October Find the values of x and y from the equations: 2x + y - 4 = 0 x - 4y + 2 = 0 A. 11/7,-5/7 B. 14/9, 8/9- C. 4/9 8/9 1y = 8 x - 4y + 2 = 0 EQUATE 142 2(4y - 2) + y - 4 = 0 Subeq. I x = 9y - 2 -> 1 x = 4(q/eta) - 2 eta y = 8/2 x = - 32/9 - 2 - 2x + 4 > 0 -> 2 x = (32 - q)/9 2y - 4 + y - 4 = 0 1y + y = 4 + 4 = - 14/9