Equation of Degree 3
Polynomial \(ax^3 + bx^2 + cx + d = 0\) has roots \(x_1\), \(x_2\) and \(x_3\), then:
\(x_1 + x_2 + x_3 = - \dfrac ba\)
\(x_1 \:.\: x_2 + x_1 \:.\: x_3 + x_2 \:.\: x_3 = \dfrac ca\)
\(x_1 \:.\: x_2 \:.\: x_3 = - \dfrac da\)
Attention to sign (−) alternating
a = (+), b = (−), c = (+), d = (−)
Example 01
Given an equation \(x^3 + 5x^2 + 8x - 4 = 0\) has roots \(x_1\), \(x_2\) and \(x_3\).
Determine:
(A) \(x_1 + x_2 + x_3\)
(B) \(x_1 \:.\: x_2 + x_1 \:.\: x_3 + x_2 \:.\: x_3\)
(C) \(x_1 \:.\: x_2 \:.\: x_3\)
\(x^3 + 5x^2 + 8x - 4 = 0\)
\(a = 1, b = 5, c = 8, d = -4\)
\(x_1 + x_2 + x_3 = - \dfrac ba\)
\(x_1 + x_2 + x_3 = - \dfrac 51\)
\(x_1 + x_2 + x_3 = \bbox[5px, border: 2px solid magenta] {- 5}\)
\(x_1 \:.\: x_2 + x_1 \:.\: x_3 + x_2 \:.\: x_3 = \dfrac ca\)
\(x_1 \:.\: x_2 + x_1 \:.\: x_3 + x_2 \:.\: x_3 = \dfrac 81\)
\(x_1 \:.\: x_2 + x_1 \:.\: x_3 + x_2 \:.\: x_3 = \bbox[5px, border: 2px solid magenta] {8}\)
\(x_1 \:.\: x_2 \:.\: x_3 = - \dfrac da\)
\(x_1 \:.\: x_2 \:.\: x_3 = - \dfrac {-4}{1}\)
\(x_1 \:.\: x_2 \:.\: x_3 = \bbox[5px, border: 2px solid magenta] {4}\)
Equation of Degree 4
Polynomial \(ax^4 + bx^3 + cx^2 + dx + e = 0\) has roots \(x_1\), \(x_2\), \(x_3\) and\(x_4\), then:
\(x_1 + x_2 + x_3 + x_4 = - \dfrac ba\)
\(x_1 \:.\: x_2 + x_1 \:.\: x_3 + x_1 \:.\: x_4 + x_2 \:.\: x_3 + x_2 \:.\: x_4 + x_3 \:.\: x_4 = \dfrac ca\)
\(x_1 \:.\: x_2 \:.\: x_3 + x_1 \:.\: x_2 \:.\: x_4 + x_2 \:.\: x_3 \:.\: x_4 = - \dfrac da\)
\(x_1 \:.\: x_2 \:.\: x_3 \:.\: x_4 = \dfrac ea\)
Attention to sign (−) alternating
a = (+), b = (−), c = (+), d = (−), e = (+)
Example 02
Given an equation \(x^4 - 2x^3 - 6x^2 - 7x - 9 = 0\) has roots \(x_1\), \(x_2\), \(x_3\) and \(x_4\).
Determine:
(A) \(x_1 + x_2 + x_3 + x_4\)
(B) \(x_1 \:.\: x_2 + x_1 \:.\: x_3 + x_1 \:.\: x_4 + x_2 \:.\: x_3 + x_2 \:.\: x_4 + x_3 \:.\: x_4\)
(C) \(x_1 \:.\: x_2 \:.\: x_3 + x_1 \:.\: x_2 \:.\: x_4 + x_2 \:.\: x_3 \:.\: x_4\)
(D) \(x_1 \:.\: x_2 \:.\: x_3 \:.\: x_4 \)
\(x^4 - 2x^3 - 6x^2 - 7x - 9 = 0\)
\(a = 1, b = -2, c = -6, d = -7, e = -9\)
\(x_1 + x_2 + x_3 + x_4 = - \dfrac ba\)
\(x_1 + x_2 + x_3 + x_4 = - \dfrac {-2}{1}\)
\(x_1 + x_2 + x_3 + x_4 = \bbox[5px, border: 2px solid magenta] {2}\)
\(x_1 \:.\: x_2 + x_1 \:.\: x_3 + x_1 \:.\: x_4 + x_2 \:.\: x_3 + x_2 \:.\: x_4 + x_3 \:.\: x_4 = \dfrac ca\)
\(x_1 \:.\: x_2 + x_1 \:.\: x_3 + x_1 \:.\: x_4 + x_2 \:.\: x_3 + x_2 \:.\: x_4 + x_3 \:.\: x_4 = \dfrac {-6}{1}\)
\(x_1 \:.\: x_2 + x_1 \:.\: x_3 + x_1 \:.\: x_4 + x_2 \:.\: x_3 + x_2 \:.\: x_4 + x_3 \:.\: x_4 = \bbox[5px, border: 2px solid magenta] {-6}\)
\(x_1 \:.\: x_2 \:.\: x_3 + x_1 \:.\: x_2 \:.\: x_4 + x_2 \:.\: x_3 \:.\: x_4 = - \dfrac da\)
\(x_1 \:.\: x_2 \:.\: x_3 + x_1 \:.\: x_2 \:.\: x_4 + x_2 \:.\: x_3 \:.\: x_4 = - \dfrac {-7}{1}\)
\(x_1 \:.\: x_2 \:.\: x_3 + x_1 \:.\: x_2 \:.\: x_4 + x_2 \:.\: x_3 \:.\: x_4 = \bbox[5px, border: 2px solid magenta] {7}\)
\(x_1 \:.\: x_2 \:.\: x_3 \:.\: x_4 = \dfrac ea\)
\(x_1 \:.\: x_2 \:.\: x_3 \:.\: x_4 = \dfrac {-9}{1}\)
\(x_1 \:.\: x_2 \:.\: x_3 \:.\: x_4 = \bbox[5px, border: 2px solid magenta] {-9}\)
Exercise