Lovely problem!

Algebra Level 2

Solve the equation $x^4-4x^3+6x^2-4x+1=0$

The answer is 1.

3 solutions

Ryan Tamburrino
Jan 2, 2015

Note the expression can be written as (x-1)^4. Therefore the only root is x = 1.

Also note that the coefficients of the expression is from Pascal's triangle :)

Marc Vince Casimiro - 6 years, 5 months ago

It would be wrong to say that the given equation has only one root.

Recall that, $\color{#EC7300}{\text{Fundamental Theorem of Algebra}}$ says, every polynomial equation of degreen $n$ in single variable has $n$ roots.

So, there are four roots, each being equal to $-1$ . One more way of saying this is that the given equation has $-1$ as its root with multiplicity $4$ .

However, there is only one solution, i.e, $-1$ , as the word $\color{#3D99F6}{\text{Solution}}$ refers to distinct roots.

Aditya Sky - 5 years ago

Noel Lo
Jun 15, 2015

Haha LOL you can simplify the equation to $(x-1)^4 = 0$

Just remember the binomial expansion.

Sunil Pradhan
Jan 5, 2015

Though it is (x + 1)^4 using factor method

sum of coefficients = 1 – 4 + 6 – 4 + 1 = 0 so (x – 1) is one of the factor. other factor can be found out by synthetic division [Here I can not show it] But other factor is

(x – 1)(x^3 – 3x^2 + 3x^2 – 1) here it is = (x – 1)(x – 1)^3 but by using factor method,

for (x^3 – 3x^2 + 3x^2 – 1) using (x – 1) as factor sum of coefficients is zero so (x – 1) is factor of it other factor can be found out by synthetic division factor is

x² – 2x + 1 = (x – 1)^2

considering all factors (x – 1)^4 then x = 1

Lovely problem!

The answer is 1.

3 solutions

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