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HSC by Topic 1995 to 2006 Polynomials Page 1 Mathematics Extension 2 HSC Examination Topic: Polynomials 06 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. HSC (i) Find the values of a and b. 3 (ii) Hence, or otherwise, express P(x) as the product of quadratic factors with real 1 coefficients. 06 4a 3 when divided by x + 1. Find a, b and c. HSC 05 The polynomial p(x) = ax3 + bx + c has a multiple zero at 1 and has remainder 4 4b Suppose α, β, γ and δ are the four roots of the polynomial equation x4 + px3 + qx2 + rx + s = 0. HSC (i) Find the values of α + β + γ + δ and α β γ + α β δ + α γ δ + β γ δ in terms 2 of p, q, r and s. (ii) (iii) Show that α 2 + β 2 + γ 2 + δ 2 = p2– 2q. 2 4 3 2 Apply the result in part (ii) to show that x – 3x + 5x + 7x – 8 = 0 cannot 1 have four real roots. (iv) By evaluating the polynomial at x = 0 and x = 1, deduce that the polynomial 2 equation x4– 3x3 + 5x2 + 7x – 8 = 0 has exactly two real roots. 05 6b Let n be an integer greater than 2. Suppose ω is an nth root of unity and ω ≠ 1. (i) HSC By expanding the left-hand side, show that 2 3 (1 + 2ω + 3ω + 4ω + · · · + nω (ii) (iii) (iv) (v) Using the 4a = z −1 prove that By expressing the left-hand side of the equation in part (iv) in terms of cos 1 1 1 π 5 3 2π π , find the exact value, in surd form, of cos . 5 5 Let α, β and γ be the zeros of the polynomial p(x) = 3x3 + 7x2 + 11x + 51. Find (i) HSC 1 ) (ω – 1) = n. , or otherwise, z2 − 1 z − z −1 1 cos θ − i sin θ = , provided that sin θ ≠ 0. cos 2θ + i sin 2θ − 1 2i sin θ 2π 2π 1 Hence, if ω = cos + i sin , find the real part of . n n ω −1 2π 4π 6π 8π 5 Deduce that 1 + 2cos + 3cos + 4cos + 5cos =- . 2 5 5 5 5 and cos 04 identity, 2 n–1 Find α2β γ + α β 2 γ + α β γ2. 2 2 1 2 (ii) Find α + β + γ . 2 (iii) Using part (ii), or otherwise, determine how many of the zeros of p(x) are 1 real. Justify your answer. 04 7b Let α be a real number and suppose that z is a complex number such that http://members.optuszoo.com.au/hscsupport/index.htm HSC by Topic 1995 to 2006 Polynomials z+ HSC (i) (i) (iii) Page 2 1 = 2cos α. z By reducing the above equation to a quadratic equation in z, solve for z 1 and use de Moivre’s theorem to show that zn + = 2cos nα. zn 1 Let w = z + . z 1 1 1 Prove that w3 + w2 – 2w – 2 = (z + ) + (z2 + ) + (z3 + ) 2 z z z3 3 Hence, or otherwise, find all solutions of cos α + cos 2α + cos 3α = 0, 3 2 in the range 0 ≤ α ≤ 2π . 03 2b Let α = −1 + i. (i) HSC (ii) (iii) 03 2d 5a 1 4 Hence, or otherwise, find a real quadratic factor of the polynomial z + 4. 5 By applying de Moivre’s theorem and by also expanding (cos θ + i sin θ ) , 2 3 Let α, β and γ be the three roots of x3 + px + q = 0, and define sn by (i) Explain why s1 = 0, and show that s2= −2p and s3= −3q. (ii) Prove that for n > 3, sn −psn−2 − qsn−3 . α2 + β 2 + γ 2 α5 + β5 + γ 5 Deduce that = 2 5 (iii) 2c 3 2 α 3 3 +β +γ 2 3 2 It is given that 2 + i is a root of P(z) = z3 + rz2 + sz + 20, where r and s are real numbers. HSC 02 Show that α is a root of the equation z + 4 = 0. sn = α n + β n + γn for n = 1, 2, 3,… HSC 02 2 4 express cos 5θ as a polynomial in cos θ . HSC 03 Express α in modulus-argument form. (i) State why 2 − i is also a root of P(z). 1 (ii) Factorise P(z) over the real numbers. 2 3 5a The equation 4x − 27x + k = 0 has a double root. Find the possible values of k. 5b Let α, β, and γ be the roots of the equation x3 − 5x2 + 5 = 0. 2 HSC 02 (i) HSC Find a polynomial equation with integer coefficients whose roots are 2 α – 1, β – 1, and γ – 1. (ii) Find a polynomial equation with integer coefficients whose roots are 2 2 2 2 α , β , and γ . (iii) 01 HSC 3b Find the value of α 3 + β 3 + γ 3. The numbers α, β and γ satisfy the equations 2 α+β+γ=3 α2 + β2 + γ2 = 1 1 1 1 + + =2 α β γ http://members.optuszoo.com.au/hscsupport/index.htm HSC by Topic 1995 to 2006 Polynomials (i) Page 3 Find the values of α β + β γ + γ α and α β γ. 3 Explain why α, β and γ are the roots of the cubic equation x3 – 3x2 + 4x – 2 = 0 (ii) 01 7b 2 Consider the equation x3 – 3x – 1 = 0, which we denote by (*). (i) HSC Find the values of α, β and γ. Let where p and q are integers having no common divisors other than +1 and 3 3 –1. Suppose that x is a root of the equation ax – 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 2 d is irrational. Assume 2 3 that r + s d is a root of (*). Show that 3r s + s d – 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 π 9 2 . (You may assume the identity cos 3θ = 4cos3 θ – 3cos θ.) 00 2b 2 Find the complex number a, given that i is a root of the equation. HSC 00 Consider the equation z2 + az + (1 + i) = 0. 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. HSC (i) (ii) Prove that α divides e. 2 4 3 2 Prove that the polynomial q(x) = 4x – x + 3x + 2x – 3 does not have an 2 integer root. 99 2d (i) HSC 99 Consider the equation 2z3 – 3z2 + 18z + 10 = 0. 4b Given that 1 – 3 i is a root of the equation, explain why 1 + 3i is another root. 3 2 2 (ii) Find all roots of the equation 2z – 3z + 18z + 10 = 0. 2 (i) Suppose the polynomial P(x) has a double root at x = α. Prove that P'(x) also 2 has a root at x = α. HSC (ii) The polynomial A(x) = x4 + ax2 + bx + 36 has a double root at x = 2. 2 Find the values of a and b. (iii) 99 5a 3 (ii) 98 2e HSC 98 4a 2 2 The roots of x + 5x + 11 = 0, are α, β and γ. (i) HSC Factorise the polynomial A(x) of part (ii) over the real numbers. Find the polynomial equation whose roots are α 2, β 2, γ 2. 2 Find the value of α + β 2 2 +γ . 1 2 3 (i) By solving the equation z + 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 (i) Suppose that k is a double root of the polynomial equation f(x). 6 7 http://members.optuszoo.com.au/hscsupport/index.htm HSC by Topic 1995 to 2006 Polynomials Page 4 Show that f ’(k) = 0. . HSC (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) 98 6a HSC Let E(x) = 1 + x + x2 x3 x4 + + . Prove E(x) has no double roots. 2 24 6 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. 2 Indicate whether each of these statements is true or false. Give reasons for your answers. 97 5c 4 7 2 P(z) = z + bz + d. The polynomial has a double root α. HSC 96 Suppose that b and d are real numbers and d ≠ 0. Consider the polynomial 5b (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 = (iv) For what values of b does P(z) have a double root equal to (v) For what values of b does P(z) have real roots? b2 . 4 3 i? Consider the polynomial equation x4 + ax3 + bx2 + cx + d = 0, 7 where a, b, c, and d are all integers. Suppose the equation has a root of the form ki, HSC where k is real, and k ≠ 0 . 95 4a (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. 4π 4π Find the least positive integer k such that cos + i sin is a solution of 7 7 (i) 4 HSC 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. 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 + t22+ 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. http://members.optuszoo.com.au/hscsupport/index.htm HSC by Topic 1995 to 2006 Polynomials A HSC Page 5 2006 3c.(i)a=3,b=1 (ii) (x2-6x+10)(x2-6x+13) 4a.a=1,b=-3,c=2 2 8 1 π 3π 5π 7π 6b.(iii)2004 4a.(i)39 (ii)-1 (iii)1 7b.(iii) , , , 2 4 4 4 4 3 9 2005 4b.(i)-p,-r 3π 2003 2b.(i) 2 cis 4 (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+50x1 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 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 ± 3i 1, (iii)1 4a.(ii)stat pt at root (iii)a=6,b=-7 1997 5c.(iv)6 (v)b<0 1995 2 4a.(i)k=7 5b.(i)0 http://members.optuszoo.com.au/hscsupport/index.htm