Understanding Engineering Mathematics

2.3.5 Equalities and identities

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A.Determine the real values ofA,B,C,Din the following identities:

(i) (s− 1 )(s+ 2 )≡As^2 +Bs+C

(ii) (x− 1 )^3 =Ax^3 + 2 Bx^2 − 3 Cx+D

(iii) (x−A)(x+B)≡x^2 − 4

(iv) (x+ 2 )(x− 3 )( 2 x− 1 )≡A(x− 1 )^2 +Bx+Cx^3 −Dx^2

(v) A(x− 1 )+B(x+ 2 )≡x− 3

(vi) (x+A)^2 +B^2 ≡x^2 − 2 x+ 5

B.Given that

A(x−a)+B(x−b)≡ax+b

determine expressions forA,Bin terms ofa,bby two different methods.

2.3.6 Roots and factors of a polynomial

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Use the factor theorem to factorise the following polynomials:

(i) u^3 −u^2 − 4 u+ 4 (ii) x^3 + 4 x^2 +x− 6

(iii) t^4 −t^3 − 7 t^2 +t+6(iv)x^3 −x^2 −x+ 1

(v) x^3 + 3 x^2 − 10 x− 24

2.3.7 Rational functions

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A.Identify the rational functions and state (a) the denominator, (b) the numerator.

(i)

x+ 1

x− 1

(ii)

x+

√

2

x^2 − 1

(iii)

√

x+ 3

x^3 − 2 x+ 1

(iv)

x^3 + 2 x− 1

x^4 − 2 x+ 3

(v)

2 x^4 − 1

x^4 + 1

B.For what values ofxdoes

1

x− 1

−

1

x− 1

exist?

2.3.8 Algebra of rational functions

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A.Put the following over a common denominator and check your results withx=0and
one other appropriate value.

(i)

2

x+ 2

−

3

x− 1

(ii)

1

x+ 3

−

2

x− 4

(iii)

2

x+ 3

−

3

x− 2

(iv)

2

x− 2

+

3

x− 3

(v)

1

x− 1

+

1

x− 2

+

1

x− 3

(vi)

3

x+ 2

−

3

x+ 3

−

4

x− 1