Theory Of Equations Sheet

THEORY OF EQUATIONS

1) Use Horner’s division to find the quotient and the remainder in dividing 3 x7 – x6 + 31 x4 + 31 x + 5 by (x + 2). 2) Given P4(x) = 2 x4 – 13 x2 + 10 x – 19 , find P4(x + 1).

3) Use Horner’s division to find the quotient and the remainder in dividing 5 x4 – 3 x3 +2 x2 – 3x + 1 by (x – 2)(2x – 3). 4) Use complete division to find the value and the derivatives of the

polynomial P4(x) = x4 + 3 x2 – 6 x + 8 at x = 2. 5) Use Horner’s division to find the quotient and the remainder in dividing by

3 x7 – x6 – 31 x4 + 31 x + 5

i) x2 – 4

ii) x2 + 3x + 2

.

6) Find the roots of the equation 2 x3 – 15 x2 + 86 x – 102 = 0 if (3 + 5i) is one of the roots. 7) Use Horner’s division to find quotient and the remainder in dividing

6 x6 – x4 +3 x2 + 2x + 1 by x3 – x .

8) Use Horner’s division to find quotient and the remainder in dividing

P3(x) = x3 + i x2 – (21 – i) x + 20 – 20 i by

(x + i -1) and hence find the roots of the equation P3(x)= 0. 9) Find the equation with roots that are the roots of the equation x3 −

1 2 1 1 x + x+ =0 2 12 18

multiplied by λ. Find the smallest

value of λ that makes all coefficients integers. 10) If a1, a2, a3 are the roots of the equation find (a ) ∑ ai2

(b) ∑ ai2 a 2j i≠ j

.

x 3 +x2 +kx +r =0

,

11) Find the equation with roots that are the roots of the equation 4 x5 – 30 x3 – 50 x – 2 = 0 with “3” added to them. 12) Find the equation with roots that are the roots of the equation

x5 + 7 x4 + 7 x3 – 8 x2 + x + 1 = 0

with opposite sign. 13) If r1, r2, r3 are the roots of the equation x3 – 3x + 1 = 0 , find the equation with roots

ri + 5 , i = 1,2,3. ri + 2

14) If “-2” is a double root of the equation P4(x) = x4 – 7 x2 + 4x + 20 = 0 , write P4 in factored form. 15) If 3 – i is a root of the equation x3 – 5 x2 + 4x + 10 = 0, find the other roots. 16) The roots of the equation x3 + p x2 + q x + r = 0 constitute a geometric series. Prove that q3 = p3 r. 17) Find the roots of the equations (i)

3 x3 + 10 x2 + x – 6 = 0

(ii)

x4 – 7x3 + 17 x2 – 17 x + 6 = 0

(iii)

x3 – 5 x 2 + 3 = 0

(iv)

x3 + x – 1 = 0

(v)

x3 – x2 – x – 1 = 0.

18) If ri, i=1,2,3,4 are the roots of the equation x4 – 2x2 + 3x + 7 = 0 find the equation with roots

2 ri2 − 1

, i = 1,2,3,4

.

19) Find the equation that has roots less by α than the roots of the equation x3 – 3x2 – 9x + 5 = 0. What is the value of α that makes the sum of the roots equals zero in the new equation?. Also, find the value of α that makes the sum of the product of pairs of roots equals zero.