by Topic 995 to 006 Polynomials Page Mathematics Etension Eamination Topic: Polynomials 06 06 05 05 c Two of the zeros of P() = + 59 8 + 0 are a + ib and a + ib, where a and b are real and b > 0. Find the values of a and b. Hence, or otherwise, epress P() as the product of quadratic factors with real coefficients. a The polynomial p() = a + b + c has a multiple zero at and has remainder when divided by +. Find a, b and c. b Suppose α, β, γ and δ are the four roots of the polynomial equation + p + q + r + s = 0. 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. Show that α + β + γ + δ = p q. (iv) Apply the result in part to show that + 5 + 7 8 = 0 cannot have four real roots. By evaluating the polynomial at = 0 and =, deduce that the polynomial equation + 5 + 7 8 = 0 has eactly two real roots. 6b Let n be an integer greater than. Suppose ω is an nth root of unity and ω. By epanding the left-hand side, show that ( + ω + ω + ω + + nω n ) (ω ) = n. z Using the identity, z =, or otherwise, prove that z z cos θ i sinθ =, provided that sin θ 0. cosθ + i sinθ i sinθ π π Hence, if ω = cos + i sin, find the real part of. n n ω π π 6π 8π 5 (iv) Deduce that + cos + cos + cos + 5cos = -. 5 5 5 5 (v) By epressing the left-hand side of the equation in part (iv) in terms of cos 5 π π π and cos, find the eact value, in surd form, of cos. 5 5 0 a Let α, β and γ be the zeros of the polynomial p() = + 7 + + 5. Find Find α β γ + α β γ + α β γ. Find α + β + γ. Using part, or otherwise, determine how many of the zeros of p() are real. Justify your answer. 0 7b Let α be a real number and suppose that z is a comple number such that
by Topic 995 to 006 Polynomials Page z + z = cos α. By reducing the above equation to a quadratic equation in z, solve for z and use de Moivre s theorem to show that z n + = cos nα. n z 0 0 0 Let w = z +. z Prove that w + w w = (z + ) + (z + ) + (z + ) z z z Hence, or otherwise, find all solutions of cos α + cos α + cos α = 0, in the range 0 α π. b Let α = + i. Epress α in modulus-argument form. Show that α is a root of the equation z + = 0. Hence, or otherwise, find a real quadratic factor of the polynomial z +. d By applying de Moivre s theorem and by also epanding (cos θ + i sin θ ) 5, epress cos 5θ as a polynomial in cos θ. 5a Let α, β and γ be the three roots of + p + q = 0, and define s n by s n = α n + β n + γ n for n =,,, Eplain why s = 0, and show that s = p and s = q. Prove that for n >, s n ps n qs n. 5 5 5 α + β + γ Deduce that = α + β + γ α + β + γ 5 0 0 c It is given that + i is a root of P(z) = z + rz + sz + 0, where r and s are real numbers. State why i is also a root of P(z). Factorise P(z) over the real numbers. 5a The equation 7 + k = 0 has a double root. Find the possible values of k. 0 0 5b Let α, β, and γ be the roots of the equation 5 + 5 = 0. Find a polynomial equation with integer coefficients whose roots are α, β, and γ. Find a polynomial equation with integer coefficients whose roots are α, β, and γ. Find the value of α + β + γ. b The numbers α, β and γ satisfy the equations α + β + γ = α + β + γ = + + = α β γ
by Topic 995 to 006 Polynomials Page 0 Find the values of α β + β γ + γ α and α β γ. Eplain why α, β and γ are the roots of the cubic equation + = 0 Find the values of α, β and γ. 7b Consider the equation = 0, which we denote by (*). Let where p and q are integers having no common divisors other than + and. Suppose that is a root of the equation a + b = 0, where a and b are integers. Eplain why p divides b and why q divides a. Deduce that (*) does not have a rational root. 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 r s + s d s = 0 and show r s d that must also be a root of (*). Deduce from this result and part, that no root of (*) can be epressed in the form r + s d with r, s and d rational. π Show that one root of (*) is cos. 9 (You may assume the identity cos θ = cos θ cos θ.) 00 b Consider the equation z + az + ( + i) = 0. 00 99 Find the comple number a, given that i is a root of the equation. 5b Consider the polynomial p() = a + b + c + d + e, where a, b, c, d and e are integers. Suppose α is an integer such that p(α) = 0. Prove that α divides e. Prove that the polynomial q() = + + does not have an integer root. d Consider the equation z z + 8z + 0 = 0. Given that i is a root of the equation, eplain why + i is another root. Find all roots of the equation z z + 8z + 0 = 0. 99 b Suppose the polynomial P() has a double root at = α. Prove that P'() also 99 98 has a root at = α. The polynomial A() = + a + b + 6 has a double root at =. Find the values of a and b. Factorise the polynomial A() of part over the real numbers. 5a The roots of + 5 + = 0, are α, β and γ. Find the polynomial equation whose roots are α, β, γ. Find the value of α + β + γ. e By solving the equation z + = 0, find the three cube roots of. 6 Let λ be a cube root of, where λ is not real. Show that λ = l λ. Hence simplify ( - λ) 6 98 a Suppose that k is a double root of the polynomial equation f(). 7
by Topic 995 to 006 Polynomials Page Show that f (k) = 0.. What feature does the graph of a polynomial have at a root of multiplicity? The polynomial P() = a 7 + b 6 + is divisible by ( ). Find the coefficients a and b. (iv) Let E() = + + + 6 +. Prove E() has no double roots. 98 97 96 95 95 6a Consider the following statements about a polynomial Q(). If Q() is even, then Q () is odd. If Q () is even, then Q() is odd. Indicate whether each of these statements is true or false. Give reasons for your answers. 5c Suppose that b and d are real numbers and d 0. Consider the polynomial P(z) = z + bz + d. The polynomial has a double root α. Prove that P (z) is an odd function. Prove that -α is also a double root of P(z). b Prove that d =. (iv) For what values of b does P(z) have a double root equal to i? (v) For what values of b does P(z) have real roots? 5b Consider the polynomial equation + a + b + c + 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. State why the conjugate -ki is also a root. Show that c = k a. Show that c + a d = abc. (iv) If is also a root of the equation, and b = 0, show that c is even. a π π Find the least positive integer k such that cos + i sin is a solution of 7 7 z k =. Show that if the comple number w is a solution of z n =, then so is w m, where m and n are arbitrary integers. 5b Let f(t) = t + ct + d, where c and d are constants. Suppose that the equation f (t) = 0 has three distinct real roots, t, t, and t. Find t + t + t. Show that t + t + t = -c. 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 7d + c < 0. 7 7
by Topic 995 to 006 Polynomials Page 5 A 006 c.a=,b= ( -6+0)( -6+) a.a=,b=-,c= 005 b.-p,-r 8 π π 5π 7π π 6b.- 00 a.9-7b.,,, 00 b. cis 9 z +z+ 00 c.(z+)(z -z+5) 5a. ± 7 5b. - -7+=0-5 +50-5=0 0 00 b.,,+i,-i 000 b.- 999 d.+i,-i,- b.a=-,b=-0 (-) ( ++9) 5a. -5-0-=0 5 ± i, a.k=7 5b.0 998 e.- a.stat pt at root a=6,b=-7 997 5c.(iv)6 (v)b<0 995