Ex. Solve x-6x2+8x-3=0. As p6, q = 8, r = − 3, 7 is found from the cubic A real root of this equation is 12 = 4, hence l = 2. Taking 7 = 2, we find from equation (3) m =- 3, n = 1, and the two quadratic equations 596. An equation is called a reciprocal equation if the reciprocal of any root is again a root. In a reciprocal equation of odd degree, one root must be its own reciprocal, hence one root must be either +1 or 1. •+an-1 Equation (3) has the same roots as equation (1). Hence the corresponding coefficients are equal or proportional. 598. Either the coefficients of terms equally distant from the first and last terms are equal or their absolute values are equal and their signs opposite. 599. A reciprocal equation of even degree having the signs of the first and last terms equal is called a standard reciprocal equation. 3 x4 - 5 x3 +7x2 - 5x + 3 = 0 is a standard reciprocal equation. 600. Any reciprocal equation can be reduced to a standard reciprocal equation by removing the factors x-1 or x+1, or both. Any reciprocal equation of odd degree has the roots + 1 or - 1 (§ 599), hence by removing the factor (x − 1) or (x + 1) it can be reduced to even degree. A reciprocal equation of even degree, with the signs of the first and last terms opposite, can have no middle term. By grouping terms it can easily be shown that x2 - 1 is a factor of the left member. By removing the factor x2 - 1, the equation is reduced to a standard equation. 601. A standard reciprocal equation can be reduced to an equation of half its dimension. Let 26-3x+5x-7x3+5x2-3x+2=0. (1) APPENDIX I. MULTIPLICATION BY DETACHED COEFFICIENTS 1. When the literal part of the product of two polynomials can be obtained by inspection, the work of multiplication may be simplified by omitting all literal factors. Ex. 1. Multiply 4x2+6x+2 by 2x2-5x-1. 2. An example containing an irregular sequence of exponents may be made regular by the insertion of zero coefficients, e.g. 23+2x+1 = x3 + 0 x2 + 2 x + 1. |