Higher Engineering Mathematics

2 NUMBER AND ALGEBRA Now try the following exercise. Exercise 1 Revision of basic operations and laws of indices Evaluate 2ab+ ...

ALGEBRA 3 A Now try the following exercise. Exercise 2 Further problems on brackets, factorization and precedence Simplify 2(p+ ...

4 NUMBER AND ALGEBRA √ R^2 +X^2 =Zand squaring both sides gives R^2 +X^2 =Z^2 , from which, X^2 =Z^2 −R^2 andreactanceX= √ Z^2 − ...

ALGEBRA 5 A and 9 x−y= 33 (6) 8 ×equation (6) gives: 72x− 8 y= 264 (7) Equation (7)−equation (5) gives: 71 x= 284 from which, x= ...

6 NUMBER AND ALGEBRA Determine the quadratic equation inxwhose roots are 2 and−5. [x^2 + 3 x− 10 =0] Solve the following quadra ...

ALGEBRA 7 A (1) (4) (7) 3 x^2 − 2 x + 5 x+ 1 ) 3 x^3 + x^2 + 3 x+ 5 3 x^3 + 3 x^2 − 2 x^2 + 3 x+ 5 − 2 x^2 − 2 x ————– 5 x+ 5 (^ ...

8 NUMBER AND ALGEBRA Now try the following exercise. Exercise 5 Further problems on polynomial division Divide (2x^2 +xy−y^2 )b ...

ALGEBRA 9 A Hence x^3 − 7 x− 6 x− 3 =x^2 + 3 x+ 2 i.e. x^3 − 7 x− 6 =(x−3)(x^2 + 3 x+2) x^2 + 3 x+2 factorizes ‘on sight’ as (x+ ...

10 NUMBER AND ALGEBRA 1.6 The remainder theorem Dividing a general quadratic expression (ax^2 +bx+c)by(x−p), wherepis any whole ...

ALGEBRA 11 A Problem 32. Determine the remainder when (x^3 − 2 x^2 − 5 x+6) is divided by (a) (x−1) and (b) (x+2). Hence factori ...

Number and Algebra 2 Inequalities 2.1 Introduction to inequalities Aninequalityis any expression involving one of the symbols< ...

INEQUALITIES 13 A Dividing both sides of the inequality: 3x>4by3 gives: x> 4 3 Hence all values ofxgreater than 4 3 satisf ...

14 NUMBER AND ALGEBRA | 3 z− 4 |>2 means 3z− 4 >2 and 3z− 4 <−2, i.e. 3z>6 and 3z<2, i.e. the inequality:| 3 z− 4 ...

INEQUALITIES 15 A It is not possible to satisfy bothx≥−1 and x<−2 thus no values ofxsatisfies (ii). Summarizing, 2 x+ 3 x+ 2 ...

16 NUMBER AND ALGEBRA From the general rule stated above in equation (2): ifx^2 <4 then− √ 4 <x< √ 4 i.e. the inequalit ...

INEQUALITIES 17 A Summarizing, t^2 − 2 t− 8 <0 is satisfied when − 2 <t< 4 Problem 17. Solve the inequality: x^2 + 6 x+ ...

Number and Algebra 3 Partial fractions 3.1 Introduction to partial fractions By algebraic addition, 1 x− 2 + 3 x+ 1 = (x+1)+3(x− ...

PARTIAL FRACTIONS 19 A whereAandBare constants to be determined, i.e. 11 − 3 x (x−1)(x+3) ≡ A(x+3)+B(x−1) (x−1)(x+3) , by algebr ...

20 NUMBER AND ALGEBRA Hence x^2 + 1 x^2 − 3 x+ 2 ≡ 1 + 3 x− 1 x^2 − 3 x+ 2 ≡ 1 + 3 x− 1 (x−1)(x−2) Let 3 x− 1 (x−1)(x−2) ≡ A (x− ...

PARTIAL FRACTIONS 21 A 3.3 Worked problems on partial fractions with repeated linear factors Problem 5. Resolve 2 x+ 3 (x−2)^2 i ...