1. ## Polynomial cubic roots

If the roots of the equation xcubed + 3xsquared - 2x +1 = 0 are e, f, g
find the value of
a) esquared(f+g) + fsquared(g+e) + gsquared(e+f)
b) esquaredfsquared + fsquaredgsquared + gquaredesquared

this knowing that e+f+g=-b/a, ef+eg+fg=c/a and efg=-d/a (general cubic equation rules)

AND

Solve the equation 4xcubed - 12xsquared +9x - 2 =0 given that two of its roots are equal

thankyou in advance for any help given ! appreciate it !

2. Originally Posted by iiharthero
Solve the equation 4xcubed - 12xsquared +9x - 2 =0 given that two of its roots are equal
$4x^3 - 12x^2 + 9x - 2 = 0$

By the rational roots theorem, the possible rational roots are
$\frac{\text{factors of}\; -2}{\text{factors of}\; 4} = \frac{\pm 1, \pm 2}{\pm 1, \pm 2, \pm 4}$
= ±1, ±2, ±1/2, ±1/4.

Start testing these factors by using the Factor Theorem or synthetic division. You'll find that x = 2 is a root:
Code:
2| 4 -12   9  -2
--     8  -8   2
-----------------
4  -4   1   0
So the polynomial factors into
\begin{aligned}
4x^3 - 12x^2 + 9x - 2 &= (x - 2)(4x^2 - 4x + 1) \\
&= (x - 2)(2x - 1)(2x - 1)
\end{aligned}

The other roots are x = 1/2 (double root).

01

3. Are you sure that $x^3+3x^2-2x+1=0$ is the right equation? Because it only has one root.

4. Hello, iiharthero!

Here's part (b) . . .

If the roots of: . $x^3 + 3x^2 - 2x +1 \:=\: 0$ . are $e, f, g$

find the value of:

$(a)\;\;e^2(f+g) + f^2(g+e) + g^2(e+f)$

$(b)\;\;e^2f^2 + f^2g^2 + g^2e^2$
(b) .We have: . $\begin{array}{ccc}
e+f+g \:=\:\text{-}3 & {\color{blue}[1]} \\ ef+fg+ge\:=\: \text{-}2 & {\color{blue}[2]} \\ efg \:=\: 1 & {\color{blue}[3]} \end{array}$

Square [2]: . $(ef+fg+ge)^2 \:=\:(\text{-}2)^2$

. . $e^2f^2 + 2ef^2g + 2e^2fg + f^2g^2 + 2fg^2e + g^2e^2 \:=\:4$

. . $(e^2f^2+f^2g^2+g^2e^2) + 2\underbrace{efg}_{1}\underbrace{(e+f+g)}_{-3} \:=\:4$

. . $(e^2f^2 + f^2g^2 + g^2e^2) + 2(1)(-3) \:=\:4$

Therefore: . $e^2f^2 + f^2g^2 + g^2e^2 \:=\:10$