Determine all the possibilities for rational roots of the polynomial $x^4 – 4x^3 + 6x^2 – 4x + 1 = 0$. Then determine how many of the real roots of the polynomial may be positive and how many may be negative. Factor the polynomial to confirm your results.
The answer is possible rational roots: $+-1$; number of possible real roots – positive: four or two or zero, negative: zero; actual roots: $x = 1, 1, 1, 1$ (a quadruple root).
Using the rational root theorem, you divide the factors of the constant, $1$, by the factors of the lead coefficient, also a 1. That step gives you only two different possibilities for rational roots: $1$ and $-1$.
The signs change four times in the original polynomial, indicating $4$ or $2$ or $0$ positive real roots. Replacing each $x$ with $-x$, you get $x^4 + 4x^3 + 6x^2 + 4x + 1 = 0$. The signs never change. The polynomial is the fourth power of the binomial $(x – 1)$, so it factors into $(x – 1)^4 = 0$, and the roots are $1, 1, 1, 1$. There are four positive roots (all the same number, of course).
Can someone explain, the factorization of the polynomial? I do not understand, how it factors into $(x – 1)^4$.
Best Answer
Two ways:
First, look at the coefficients: $1, 4, 6, 4, 1$ with alternating signs. If you know the binomial theorem, you can easily associate this to the expansion of $(x-1)^4$. You'd have to be pretty darn clever.
Second, you can just find the roots via synthetic division. You'll see that you'll have $x=1$ as the only root with multiplicity $4$. This corresponds to the factors $(x-1)(x-1)(x-1)(x-1)$.
You can even check that $(x-1)^4$ yields the above expansion by multiplying.