Patterns of coefficients.

Question

I am working through the book Mathematical Methods for Physics and Engineering and I came across

Beneath the question, they stated that the root x=-1 was found from the pattern of coefficients. I cannot find anything about this online and so I am very much stumped. Would appreciate if anyone could shed any light on this method of finding a root.

Many thanks.

Answer 1

The possible integer roots of $f(x)=3x^4-x^3-10x^2-2x+4=0$ are the divisors of $4.$ That is $\pm 1,\pm 2,\pm 4.$ It easy to get that

$$f(1)=3-1-10-2+4\ne 0,$$ which shows that $x=1$ is not a root, and $$f(-1)=3+1-10+2+4=0,$$ which shows that $x=-1$ is a root. In other words, the sum of the coefficients is not zero and thus $x=1$ is not a root. And the sum of the coefficients of even degree minus the sum of the coefficients of odd degree is zero and thus $x=-1$ is a root.

Now, the way to factor $f(x)$ is not the shortest nor the simplest. Since $x=-1$ is a root we have that

$$3x^4-x^3-10x^2-2x+4=(x+1)(b_3x^3+b_2x^2+b_1x+b_0).$$ Identifying the coefficients of $x^4$ we have $$b_3=3;$$ identifying the coefficients of $x^3$ we have $$b_2+b_3=-1,$$ and so on.

If we use Ruffini's rule (see https://en.wikipedia.org/wiki/Ruffini%27s_rule) we will get the answer quickly. Or just make the division.

Answer 2

I think what they mean is simply this: Begin by supposing that -1 could be a root. Plug -1 into the expression. We get:

3 - (-1) - 10 - (-2) + 4 = 3 + 1 - 10 + 2 + 4 = 0

which confirms our supposition.

More generally, it is a good idea to try plugging a few small-magnitude integers into the algebraic expression because the result is a relatively simple arithmetic expression. Values 0, 1, and -1 in particular are good choices because their powers are trivial to evaluate. If the polynomial is monic, of course, one tries factors of the constant term.