1.9 HINTS AND ANSWERS
1.27 Establish the values ofkfor which the binomial coefficientpCkis divisible byp
whenpis a prime number. Use your result and the method of induction to prove
thatnp−nis divisible bypfor all integersnand all prime numbersp. Deduce
thatn^5 −nisdivisibleby30foranyintegern.
1.28 An arithmetic progression of integersanis one in whichan=a 0 +nd,wherea 0
anddare integers andntakes successive values 0, 1 , 2 ,....
(a) Show that if any one term of the progression is the cube of an integer then
so are infinitely many others.
(b) Show that no cube of an integer can be expressed as 7n+ 5 for some positive
integern.1.29 Prove, by the method of contradiction, that the equation
xn+an− 1 xn−^1 +···+a 1 x+a 0 =0,in which all the coefficientsaiare integers, cannot have a rational root, unless
that root is an integer. Deduce that any integral root must be a divisor ofa 0 and
hence find all rational roots of(a) x^4 +6x^3 +4x^2 +5x+4=0,
(b)x^4 +5x^3 +2x^2 − 10 x+6=0.Necessary and sufficient conditions1.30 Prove that the equationax^2 +bx+c=0,inwhicha,bandcare real anda>0,
has two real distinct solutions IFFb^2 > 4 ac.
1.31 For the real variablex, show that a sufficient, but notnecessary, condition for
f(x)=x(x+ 1)(2x+ 1) to be divisible by 6 is thatxis an integer.
1.32 Given that at least one ofaandb, and at least one ofcandd, are non-zero,
show thatad=bcis both a necessary and sufficient condition for the equations
ax+by=0,
cx+dy=0,
to have a solution in which at least one ofxandyis non-zero.
1.33 The coefficientsaiin the polynomialQ(x)=a 4 x^4 +a 3 x^3 +a 2 x^2 +a 1 xare all
integers. Show thatQ(n) is divisible by 24 for all integersn≥0 if and only if all
of the following conditions are satisfied:
(i) 2a 4 +a 3 is divisible by 4;
(ii)a 4 +a 2 is divisible by 12;
(iii)a 4 +a 3 +a 2 +a 1 is divisible by 24.
1.9 Hints and answers1.1 (b) The roots are 1,^18 (−7+
√
33) =− 0. 1569 ,^18 (− 7 −
√
33) =− 1 .593. (c)−5and
7
4 are the values ofkthat makef(−1) andf(1
2 ) equal to zero.
1.3 (a) a=4,b=^38 andc=^2316 are all positive. Thereforef′(x)>0forallx>0.
(b)f(1) = 5,f(0) =−2andf(−1) = 5, and so there is at least one root in each
of the ranges 0<x<1and− 1 <x<0. (x^7 +5x^6 )+(x^4 −x^3 )+(x^2 −2)
is positive definite for− 5 <x<−
√
- There are therefore no roots in this
range, but there must be one to the left ofx=−5.
1.5 (a)x^2 +9x+18=0; (b)x^2 − 4 x=0;(c)x^2 − 4 x+4=0; (d)x^2 − 6 x+13=0.
1.7 (a) Use sin(π/4) = 1/
√
- (b) Use results (1.20) and (1.21).
1.9 (a) 1.339,− 2 .626. (b) No solution because 6^2 > 42 +3^2 .(c)− 0 .0849,− 2 .276.