Math Help - Roots of polynomial equations: Substitution method.

1. Roots of polynomial equations: Substitution method.

How would you do this one with the substitution method?
Equation
$x^2 + 4x + 7$
Find an equation with roots $\alpha$ + $2\beta$ and $\beta$ + $2\alpha$

Example of what i mean:
$x^2 + 5x + 7 = 0$ has roots $\alpha$ and $\beta$
Find an equation with $2\alpha$ and $2\beta$
Substitution would be:
Let u = $2\alpha$
hence $\alpha$ = $\frac{u}{2}$

Sub it into equation gives.
$(\frac{u}{2})^2 + 5(\frac{u}{2}) + 7$
When multiplied out gives
$u^2 + 10u + 28$

Uh, i don't mean to sound patronising when i gave an example since i know a lot of you are muuuch better at maths then me . Just wanted to make sure my question was clear, thats all .

Thankyou very much

2. Originally Posted by AshleyT
Given that $\alpha$ and $\beta$ are roots of the equation $x^2 + 4x + 7 = 0$. Find an equation with roots $\alpha+2\beta$ and $\beta + 2\alpha$
Hello Ashley, A lengthy example is not required you just need to clearly define your question.

Given the root of your quadratic you can write. $x^2 + 4x + 7 = (x -\alpha)(x-\beta)$

Expand and compare coefficients to give.
$\alpha+\beta = -4 \ \ \ (1)$
$\alpha \beta = 7 \ \ \ \ \ \ \ \ (2)$

now you require a quadratic such that $\alpha+2\beta$ and $\beta + 2\alpha$ are roots. So your required quadratic can be written as $(x -( \alpha+2\beta))(x-(\beta + 2\alpha))$

$\Rightarrow x^2 -3(\alpha+\beta)x + \alpha \beta +2 \beta^2 + 4 \alpha \beta + 2 \alpha^2$

$\Rightarrow x^2 -3(\alpha+\beta)x + \alpha \beta +2(\beta + \alpha)^2$

Bobak

3. Originally Posted by bobak
Hello Ashley, A lengthy example is not required you just need to clearly define your question.

Given the root of your quadratic you can write. $x^2 + 4x + 7 = (x -\alpha)(x-\beta)$

Expand and compare coefficients to give.
$\alpha+\beta = -4 \ \ \ (1)$
$\alpha \beta = 7 \ \ \ \ \ \ \ \ (2)$

now you require a quadratic such that $\alpha+2\beta$ and $\beta + 2\alpha$ are roots. So your required quadratic can be written as $(x -( \alpha+2\beta))(x-(\beta + 2\alpha))$

$\Rightarrow x^2 -3(\alpha+\beta)x + \alpha \beta +2 \beta^2 + 4 \alpha \beta + 2 \alpha^2$

$\Rightarrow x^2 -3(\alpha+\beta)x + \alpha \beta +2(\beta + \alpha)^2$