Page:Elementary algebra (1896).djvu/486

468 Rh EXAMPLES XLVIII. d.

1. Transform the equation rc^ — Sx* + Oa:^ — 9.x2 -f ox — 1 = into another which shall have the same roots with contrary signs. 2. Transform the equation 2x^ — 4:X^ + 7x — 3 = into another whose roots shall be those of the first multiplied by 3. 3. Transform the equation x^ — 7x — 6 = into another whose roots shall be those of the first multiplied by — |. 4. Transform the equation x^ — 4^^ + ix — ^ = into another with integral coefficients, and unity for the coefficient of the first term. 5. Transform the equation 3 x* — 5x3 + x^ — x + 1 = into another the coefficient of whose first term is unity. 6. Transform the equation x^ + 10 x^ -f 39 x^ + 76 x + 65 = into another whose roots shall be greater by 4. 7. Transform the equation x* — 12 x^ + 17 x'^ — 9x + 7 = into another whose roots shall be less by 3. 8. Diminish by 1 the roots of the equation 2x4- 13x2 + 10x- 19 = 0. 9. Find the equation whose roots are greater by 4 than the corresponding roots of x4 + 16 x3 + 72 x^ + 64 x - 129 = 0. 10. Solve the equation 3 x^ - 22 x^ + 48 x - 32 = 0, the roots of which are in harmonical progression. 11. The roots of x^ — 11 x'^ + 36x — 36 = are in harmonical progression ; find them.

Remove the second term from the equations :

12. x3-6x2 + 10x-3 = 0. 13. x4 + 4 x3 + 2 X- - 4 X - 2 = 0. X 3

14. Transform the equation $$x^3 - \frac{x}{4}-\frac{3}{4}=0$$ into one whose roots exceed by $$\frac{3}{2}$$ the corresponding roots of the given equation.

15. Diminish by 3 the roots of the equation $$x^5-4x^4+3x^2-4x+6=0.$$

16. Find the equation each of whose roots is greater by unity than a root of the equation $$x^3-5x^2+6x-3=0.$$

17. Find the equation whose roots are the squares of the roots of $$x^4+x^3+2x^2+x+1=0.$$

18. Form the equation whose roots are the cubes of the roots of $$x^3+3x^2+2=0.$$