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Mathematics Extension 2 HSC Examination

Topic: Polynomials

06

HSC

3c Two of the zeros of P(x) = x4– 12x3 + 59x2 – 138x + 130 are a + ib and a + 2ib,

where a and b are real and b > 0.

(i) Find the values of a and b.

(ii) Hence, or otherwise, express P(x) as the product of quadratic factors with real

coefficients.

3

1

06

HSC

4a The polynomial p(x) = ax3 + bx + c has a multiple zero at 1 and has remainder 4

when divided by x + 1. Find a, b and c.

3

05

HSC

4b Suppose α, β, γ and δ are the four roots of the polynomial equation

x4 + px3 + qx2 + rx + s = 0.

(i) Find the values of α + β + γ + δ and α β γ + α β δ + α γ δ + β γ δ in terms

of p, q, r and s.

(ii) Show that α 2 + β 2 + γ 2 + δ 2 = p2– 2q.

(iii) Apply the result in part (ii) to show that x4– 3x3 + 5x2 + 7x – 8 = 0 cannot

have four real roots.

(iv) By evaluating the polynomial at x = 0 and x = 1, deduce that the polynomial

equation x4– 3x3 + 5x2 + 7x – 8 = 0 has exactly two real roots.

2

2

1

2

05

HSC

6b Let n be an integer greater than 2. Suppose ω is an nth root of unity and ω ≠ 1.

(i) By expanding the left-hand side, show that

(1 + 2ω + 3ω2 + 4ω3 + · · · + nωn–1) (ω – 1) = n.

(ii) Using the identity, 1

12 −z

= 1

1

−

−

− zz

z, or otherwise, prove that

12sin2cos

1

−+ θθ i =

θ

θθ

sin2

sincos

i

i−, provided that sin θ ≠ 0.

(iii) Hence, if ω = cos n

π2 + i sin

n

π2, find the real part of

1

1

−ω.

(iv) Deduce that 1 + 2cos5

2π + 3cos

5

4π + 4cos

5

6π + 5cos

5

8π = -

2

5.

(v) By expressing the left-hand side of the equation in part (iv) in terms of cos5

π

and cos5

2π, find the exact value, in surd form, of cos

5

π.

2

1

1

1

3

04

HSC

4a Let α, β and γ be the zeros of the polynomial p(x) = 3x3 + 7x2 + 11x + 51. Find

(i) Find α2β γ + α β2 γ + α β γ2.

(ii) Find α2 + β2 + γ2.

(iii) Using part (ii), or otherwise, determine how many of the zeros of p(x) are

real. Justify your answer.

1

2

1

04 7b Let α be a real number and suppose that z is a complex number such that

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HSC z + z

1 = 2cos α.

(i) By reducing the above equation to a quadratic equation in z, solve for z

and use de Moivre’s theorem to show that zn + nz

1 = 2cos nα.

(i) Let w = z + z

1.

Prove that w3 + w2 – 2w – 2 = (z + z

1) + (z2 +

2

1

z) + (z3 +

3

1

z)

(iii) Hence, or otherwise, find all solutions of cos α + cos 2α + cos 3α = 0,

in the range 0 ≤ α ≤ 2π .

3

2

3

03

HSC

2b Let α = −1 + i.

(i) Express α in modulus-argument form.

(ii) Show that α is a root of the equation z4 + 4 = 0.

(iii) Hence, or otherwise, find a real quadratic factor of the polynomial z4 + 4.

2

1

2

03

HSC

2d By applying de Moivre’s theorem and by also expanding (cos θ + i sin θ )5,

express cos 5θ as a polynomial in cos θ .

3

03

HSC

5a Let α, β and γ be the three roots of x3 + px + q = 0, and define sn by

sn = α n + β n + γn for n = 1, 2, 3,…

(i) Explain why s1 = 0, and show that s2= −2p and s3= −3q.

(ii) Prove that for n > 3, sn −psn−2 − qsn−3 .

(iii) Deduce that 5

555 γβα ++ =

++

2

222 γβα

++

2

333 γβα

3

2

2

02

HSC

2c It is given that 2 + i is a root of P(z) = z3 + rz2 + sz + 20, where r and s are real

numbers.

(i) State why 2 − i is also a root of P(z).

(ii) Factorise P(z) over the real numbers.

1

2

02

HSC

5a The equation 4x3 − 27x + k = 0 has a double root. Find the possible values of k.

2

02

HSC

5b Let α, β, and γ be the roots of the equation x3 − 5x2 + 5 = 0.

(i) Find a polynomial equation with integer coefficients whose roots are

α – 1, β – 1, and γ – 1.

(ii) Find a polynomial equation with integer coefficients whose roots are

α 2, β 2, and γ 2.

(iii) Find the value of α 3 + β 3 + γ 3.

2

2

2

01

HSC

3b The numbers α, β and γ satisfy the equations α + β + γ = 3

α 2 + β 2 + γ 2 = 1

α

1 +

β

1 +

γ

1 = 2

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(i) Find the values of α β + β γ + γ α and α β γ.

Explain why α, β and γ are the roots of the cubic equation x3 – 3x2 + 4x – 2 = 0

(ii) Find the values of α, β and γ.

3

2

01

HSC

7b Consider the equation x3 – 3x – 1 = 0, which we denote by (*).

(i) Let where p and q are integers having no common divisors other than +1 and

–1. Suppose that x is a root of the equation ax3 – 3x + b = 0, where a and b

are integers. Explain why p divides b and why q divides a. Deduce that (*)

does not have a rational root.

(ii) Suppose that r, s and d are rational numbers and that d is irrational. Assume

that r + s d is a root of (*). Show that 3r2s + s3d – 3s = 0 and show

r – s d that must also be a root of (*). Deduce from this result and part (i),

that no root of (*) can be expressed in the form r + s d with r, s and d

rational.

(iii) Show that one root of (*) is 2 cos 9

π.

(You may assume the identity cos 3θ = 4cos3 θ – 3cos θ.)

3

2

2

00

HSC

2b Consider the equation z2 + az + (1 + i) = 0.

Find the complex number a, given that i is a root of the equation.

2

00

HSC

5b Consider the polynomial p(x) = ax4 + bx3 + cx2 + dx + e, where a, b, c, d and e are

integers. Suppose α is an integer such that p(α) = 0.

(i) Prove that α divides e.

(ii) Prove that the polynomial q(x) = 4x4 – x3 + 3x2 + 2x – 3 does not have an

integer root.

2

2

99

HSC

2d Consider the equation 2z3 – 3z2 + 18z + 10 = 0.

(i) Given that 1 – 3 i is a root of the equation, explain why 1 + 3i is another root.

(ii) Find all roots of the equation 2z3 – 3z2 + 18z + 10 = 0.

2

2

99

HSC

4b (i) Suppose the polynomial P(x) has a double root at x = α. Prove that P'(x) also

has a root at x = α.

(ii) The polynomial A(x) = x4 + ax2 + bx + 36 has a double root at x = 2.

Find the values of a and b.

(iii) Factorise the polynomial A(x) of part (ii) over the real numbers.

2

2

2

99

HSC

5a The roots of x3 + 5x2 + 11 = 0, are α, β and γ.

(i) Find the polynomial equation whose roots are α 2, β 2, γ 2.

(ii) Find the value of α 2 + β 2 + γ 2.

1

2

98

HSC

2e (i) By solving the equation z3 + 1 = 0, find the three cube roots of –1.

(ii) Let λ be a cube root of –1, where λ is not real. Show that λ2= l – λ.

(iii) Hence simplify (1 - λ)6

6

98 4a (i) Suppose that k is a double root of the polynomial equation f(x). 7

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HSC Show that f ’(k) = 0. .

(ii) What feature does the graph of a polynomial have at a root of multiplicity 2?

(iii) The polynomial P(x) = ax7 + bx6 + 1 is divisible by (x – 1)2. Find the

coefficients a and b.

(iv) Let E(x) = 1 + x + 2

2x +

6

3x +

24

4x. Prove E(x) has no double roots.

98

HSC

6a Consider the following statements about a polynomial Q(x).

(i) If Q(x) is even, then Q’(x) is odd.

(ii) If Q’(x) is even, then Q(x) is odd.

Indicate whether each of these statements is true or false. Give reasons for your

answers.

2

97

HSC

5c Suppose that b and d are real numbers and d ≠ 0. Consider the polynomial

P(z) = z4 + bz2 + d. The polynomial has a double root α.

(i) Prove that P’(z) is an odd function.

(ii) Prove that -α is also a double root of P(z).

(iii) Prove that d = 4

2b.

(iv) For what values of b does P(z) have a double root equal to 3 i?

(v) For what values of b does P(z) have real roots?

7

96

HSC

5b Consider the polynomial equation x4 + ax3 + bx2 + cx + d = 0,

where a, b, c, and d are all integers. Suppose the equation has a root of the form ki,

where k is real, and k ≠ 0 .

(i) State why the conjugate -ki is also a root.

(ii) Show that c = k2a.

(iii) Show that c2 + a2d = abc.

(iv) If 2 is also a root of the equation, and b = 0, show that c is even.

7

95

HSC

4a (i) Find the least positive integer k such that cos

7

4π + i sin

7

4π is a solution of

zk = 1.

(ii) Show that if the complex number w is a solution of zn = 1, then so is wm,

where m and n are arbitrary integers.

4

95

HSC

5b Let f(t) = t3 + ct + d, where c and d are constants. Suppose that the equation

f (t) = 0 has three distinct real roots, t1, t2, and t3.

(i) Find t1 + t2 + t3.

(ii) Show that t12 + t2

2+ t32 = -2c.

(iii) Since the roots are real and distinct, the graph of y=f(t) has two turning points,

at t = u and t = v, and f(u).f(v)<0.

Show that 27d2 + 4c3 < 0.

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A

HSC

2006 3c.(i)a=3,b=1 (ii) (x2-6x+10)(x2-6x+13) 4a.a=1,b=-3,c=2 2005 4b.(i)-p,-r

6b.(iii)-2

1 2004 4a.(i)39

3

2 (ii)-1

9

8 (iii)1 7b.(iii)

4

π,

4

3π,4

5π,4

7π 2003 2b.(i) 2 cis

4

3π

(iii)z2+2z+2 2002 2c.(ii)(z+4)(z2-4z+5) 5a. 27± 5b.(i)x 3-2x 2-7x+1=0 (ii)x3-25x2+50x-

25=0 (iii)110 2001 3b.(i)4,2 (ii)1,1+i,1-i 2000 2b.-1 1999 2d.(ii)1+3i,1-3i,-2

1

4b.(ii)a=-3,b=-20 (iii)(x-2)2(x2+4x+9) 5a.(i)x3-25x2-110x-121=0 (ii)25 1998 2e.(i)-

1,2

31 i± (iii)1 4a.(ii)stat pt at root (iii)a=6,b=-7 1997 5c.(iv)6 (v)b<0 1995

4a.(i)k=7 5b.(i)0