simultaneous equations with surds and polynomials!

**mathmo** · February 22nd 2009, 02:46 AM

1. solve the following simultaneous equations, leaving your answers in surd form:

(x+yroot2)^2 = 27 + 12root2 and y = 2x

2. (x-A) is a factor of f(x) where f(x) - x^3 + x^2 - x + A
a) calculate the possible values of A

b) Given that A<0 solve the equation f(x) = 0

**red_dog** · February 22nd 2009, 04:38 AM

1) Replace y in the first equation:

$(x+2x\sqrt{2})^2=27+12\sqrt{2}\Leftrightarrow x^2(9+4\sqrt{2})=27+12\sqrt{2}\Leftrightarrow$

$\Leftrightarrow x^2=\frac{27+12\sqrt{2}}{9+4\sqrt{2}}=\frac{3(9+4\ sqrt{2}}{9+\sqrt{2}}=3\Rightarrow x=\pm\sqrt{3}\Rightarrow y=\pm 2\sqrt{3}$

The solutions are: $\left\{\begin{array}{ll}x=\sqrt{3}\\y=2\sqrt{3}\en d{array}\right., \ \left\{\begin{array}{ll}x=-\sqrt{3}\\y=-2\sqrt{3}\end{array}\right.$

2) If $x-A$ is a factor then $f(A)=0\Rightarrow A^3+A^2-A+A=0\Rightarrow A^2(A+1)=0\Rightarrow A=0, \ A=-1$

For $A=-1\Rightarrow f(x)=x^3+x^2-x-1=x^2(x+1)-(x+1)=(x+1)(x^2-1)=(x+1)^2(x-1)$

$f(x)=0\Rightarrow x_1=x_2=-1, \ x_3=1$

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