# Roots of Polynomial Equations

### Relation between roots and coefficients

###### 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)   $$x_1 \:.\: x_2 \:.\: x_3 \:.\: x_4$$

$$x^4 - 2x^3 - 6x^2 - 7x - 9 = 0$$

$$a = 1, b = -2, c = -6, d = -7, e = -9$$

$$x_1 + x_2 + x_3 + x_4 = - \dfrac ba$$

$$x_1 + x_2 + x_3 + x_4 = - \dfrac {-2}{1}$$

$$x_1 + x_2 + x_3 + x_4 = \bbox[5px, border: 2px solid magenta] {2}$$

$$x_1 \:.\: x_2 + x_1 \:.\: x_3 + x_1 \:.\: x_4 + x_2 \:.\: x_3 + x_2 \:.\: x_4 + x_3 \:.\: x_4 = \dfrac ca$$

$$x_1 \:.\: x_2 + x_1 \:.\: x_3 + x_1 \:.\: x_4 + x_2 \:.\: x_3 + x_2 \:.\: x_4 + x_3 \:.\: x_4 = \dfrac {-6}{1}$$

$$x_1 \:.\: x_2 + x_1 \:.\: x_3 + x_1 \:.\: x_4 + x_2 \:.\: x_3 + x_2 \:.\: x_4 + x_3 \:.\: x_4 = \bbox[5px, border: 2px solid magenta] {-6}$$

$$x_1 \:.\: x_2 \:.\: x_3 + x_1 \:.\: x_2 \:.\: x_4 + x_2 \:.\: x_3 \:.\: x_4 = - \dfrac da$$

$$x_1 \:.\: x_2 \:.\: x_3 + x_1 \:.\: x_2 \:.\: x_4 + x_2 \:.\: x_3 \:.\: x_4 = - \dfrac {-7}{1}$$

$$x_1 \:.\: x_2 \:.\: x_3 + x_1 \:.\: x_2 \:.\: x_4 + x_2 \:.\: x_3 \:.\: x_4 = \bbox[5px, border: 2px solid magenta] {7}$$

$$x_1 \:.\: x_2 \:.\: x_3 \:.\: x_4 = \dfrac ea$$

$$x_1 \:.\: x_2 \:.\: x_3 \:.\: x_4 = \dfrac {-9}{1}$$

$$x_1 \:.\: x_2 \:.\: x_3 \:.\: x_4 = \bbox[5px, border: 2px solid magenta] {-9}$$