MHB Question about roots/synthetic division

[Madds]

View attachment 7376
According to the synthetic division done below, what was the original polynomial and what number do we know is a root of that polynomial? Explain how you know to receive full points.

I know what the polynomial is, but I thought the root of this polynomial would be 1 but it's actually -1 could somebody explain why?

 

[MarkFL]

When doing synthetic division, the actual root $r$ or zero of the polynomial is put out to the left. Thus we know:

$$f(r)=0$$

And the polynomial function will contain the factor $(x-r)$. This might be why you felt the number needed to be negated?

 

[HallsofIvy]

Your question is puzzling. Are you clear on what "synthetic division" is? It is a quick way of dividing a polynomial by x- a for some value of a. Specifically, we write just the coefficients of the polynomial (here that is "5 6 7 6" which tells us that the original polynomial was [math]5x^3+ 6x^2+ 7x+ 6[/math]. The number we are "dividing" by is a= -1. In general, if we divide a polynomial, P(x), by x- a, the remainder is P(a).

The synthetic division here, shows that when x, in this polynomial, is set to -1, the value of the polynomial. is 0. That means that x= -1 is a root.