Understanding Engineering Mathematics

B.Expand the following expressions, collecting like terms:

(i) (x− 1 )(x+ 2 ) (ii) (x− 1 )(x+ 2 )(x+ 4 )

(iii) (x− 1 )(x+ 1 )^2 (iv) (x− 2 )(x− 3 )(x+ 1 )(x+ 2 )

(v) (u− 1 )^2 (u+ 1 )^2 (vi) (x− 1 )^3 (x+ 2 )

(vii) (t+ 1 )(t− 2 )(t+ 2 ) (viii) (u− 2 )(u+ 3 )(u− 3 )

(ix) (s− 2 )^4 (x) (x− 1 )(x+ 2 )(x− 3 )(x+ 4 )

(xi) (x+ 2 )^2 (x− 3 )^2 (xii) ( 2 t+ 1 )( 3 t− 4 )

(xiii) ( 3 s− 1 )(s+ 2 )( 4 s+ 3 ) (xiv) ( 3 x+ 2 )^2 ( 2 x− 1 )(x+ 2 )

(xv) ( 3 x+ 1 )( 3 x− 1 )(x+ 3 )

Check each expansion with suitable numerical values.

2.3.3 Factorisation of polynomials by inspection

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A.Factorise the following, retaining only real coefficients:

(i) x^2 +x (ii) 3x^3 − 2 x^2 (iii) − 7 x^2 + 42 x^4

(iv) t^4 − 3 t^3 +t^2 (v) u^2 −9(vi)t^2 − 121

(vii) s^24 − 16 s^22 (viii) 4x^12 − 64 x^8

B.Factorise the following polynomial expressions (hint: look back at Q2.3.2B):

(i) t^2 + 5 t+ 6 (ii) t^3 +t^2 − 4 t− 4 (iii) y^4 − 2 y^3 − 7 y^2 + 8 y+ 12

(iv) 9x^3 + 27 x^2 −x− 3

2.3.4 Simultaneous equations

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A.Solve the following systems of linear equations, verifying your solution by back substi-
tution in each case.

(i) x−y= 1 (ii) A+B= 0 (iii) s+ 3 t= 1

x+ 2 y= 03 A−B= 1 s− 2 t= 1

(iv) 3x+ 2 y=2(v)u+ 4 v=1(vi)7x 1 − 2 x 2 = 1

− 2 x+ 3 y= 1 u−v= 23 x 1 − 2 x 2 = 0

B.Comment on the following systems of equations:

(i) x+y= 1 (ii) 2x−y= 3 (iii) x+y= 0

3 x+ 3 y= 34 x− 2 y= 1 x−y= 0

(iv) 2A+B=1(v)u+v=−1(vi)x+y= 0

4 A+ 2 B=− 13 u+ 3 v=− 3 x^2 −y^2 = 1