Understanding Engineering Mathematics

D. (i) 2x+ 4 (ii) 2x+ 1 (iii) 3t^2 − 3 t

(iv) s^2 +st− 2 t^2 (v) a^3 − 3 a^2 (vi) x^3 +x^2 − 3 x+ 1

(vii) − 2 u^3 − 6 u (viii) 9x^2 − 2 x− 35 (ix) a^3 + 2 a^2 − 4 a+ 7

(x) x^3 − 2 x^2 − 2 x (xi) x^3 − 3 x^2 + 2 x (xii) −x^3 + 3 x^2 − 8

(xiii) 1−s−t−u+st+su+tu−sut (xiv) − 8 ab

(xv) 2x^2 + 2 y^2

E. In cases (i), (iii), (iv), (v), (vi), (vii), (xi), (xiii) you should retrieve the original question.
(ii), (viii), (ix) do not factorise further with simple integer roots. (x) becomesx(x^2 −
2 x− 2 ), (xii) can be written−(x^3 − 3 x^2 + 8 )but does not easily factorise further.
(xiv) is already factorised! (xv) can be written 2(x^2 +y^2 ).

2.3.2 Polynomials

A.
Polynomial? Degree Coefficients

(i) Yes 2 1,−1, 4

(ii) Yes 0 0

(iii) No

(iv) Yes 3 7, 0,−2, 1

(v) No

(vi) Yes 4 27, 0,−3, 0, 1

(vii) No

(viii) No

(ix) Yes (inxandt) 3 Coefficients all zero

except ofx^2 ( 3 )andt^3 ( 1 )

(x) No

B. (i) x^2 +x− 2 (ii) x^3 + 5 x^2 + 2 x− 8

(iii) x^3 +x^2 −x−1(iv)x^4 − 2 x^3 − 7 x^2 + 8 x+ 12

(v) u^4 − 2 u^2 +1(vi)x^4 −x^3 − 3 x^2 + 5 x− 2

(vii) t^3 +t^2 − 4 t− 4 (viii) u^3 − 2 u^2 − 9 u+ 18

(ix) s^4 − 8 s^3 + 24 s^2 − 32 s+ 16 (x) x^4 + 2 x^3 − 13 x^2 − 14 x+ 24

(xi) x^4 − 2 x^3 − 11 x^2 + 12 x+ 36 (xii) 6t^2 − 5 t− 4

(xiii) 12s^3 + 29 s^2 + 7 s− 6 (xiv) 9x^3 + 27 x^2 −x− 3

2.3.3 Factorisation of polynomials by inspection

A. (i) x(x+ 1 ) (ii) x^2 ( 3 x− 2 ) (iii) 7x^2 (

√

6 x− 1 )(

√

6 x+ 1 )

(iv) t^2 (t^2 − 3 t+ 1 ) (v) (u− 3 )(u+ 3 ) (vi) (t− 11 )(t+ 11 )

(vii) s^22 (s− 4 )(s+ 4 ) (viii) 4x^8 (x− 2 )(x+ 2 )(x^2 + 4 )