Solve the following equation $x^4 - 10x^3 + 26x^2 - 10x + 1 = 0$.

Solve the following equation x⁴ - 10x³ + 26x² - 10 – Class 12 Mathematics

Question 30 of 36

x^4 - 10x^3 + 26x^2 - 10x + 1 = 0

Telangana TsbieTelangana Board of Intermediate Education 2024Subjective· 7mImportance★★★★★

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This is a reciprocal (palindromic) equation — divide by $x^2$ and substitute $y=x+\frac1x$ to reduce it to a quadratic.

The coefficients $1,-10,26,-10,1$ are palindromic, so $x^4-10x^3+26x^2-10x+1=0$ is a reciprocal equation of even degree (type I).

Divide throughout by $x^2$ (note $x=0$ is not a root):

$x^2-10x+26-\dfrac{10}{x}+\dfrac{1}{x^2}=0$

$\left(x^2+\dfrac{1}{x^2}\right) - 10\left(x+\dfrac{1}{x}\right)+26=0$

Let $y=x+\dfrac1x$ , so $x^2+\dfrac{1}{x^2}=y^2-2$ :

$(y^2-2)-10y+26=0 \Rightarrow y^2-10y+24=0 \Rightarrow (y-4)(y-6)=0$

So $y=4$ or $y=6$ .

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