Étudier le signe du polynôme f(x) = -5x + 6x - 7

The given polynomial is $f(x) = -5x + 6x - 7$.
Combine the terms with $x$:

$$f(x) = (-5+6)x - 7$$ $$f(x) = x - 7$$

$$x - 7 = 0$$ $$x = 7$$

Since $f(x) = x - 7$ is a linear function with a positive slope (coefficient of $x$ is $1 > 0$), the function is increasing.

• For $x < 7$, $f(x)$ will be negative.
• For $x = 7$, $f(x)$ will be zero.
• For $x > 7$, $f(x)$ will be positive.

We can summarize this in a sign table:

$$\begin{array}{|c|ccccccc|} \hline x & -\infty & & 7 & & +\infty \\ \hline f(x) & & - & 0 & + & \\ \hline \end{array}$$

The sign of the polynomial $f(x) = x-7$ is:

• $f(x) < 0$ for $x \in (-\infty, 7)$
• $f(x) = 0$ for $x = 7$
• $f(x) > 0$ for $x \in (7, +\infty)$