$$-x^3+21x^2-147x+343 = 0$$
Answer
Explanation
$ \color{blue}{ -x^{3}+21x^{2}-147x+343 } $ is a polynomial of degree 3. To find zeros for polynomials of degree 3 or higher we use Rational Root Test.
The Rational Root Theorem tells you that if the polynomial has a rational zero then it must be a fraction $ \dfrac{p}{q} $, where p is a factor of the trailing constant and q is a factor of the leading coefficient.
The factor of the leading coefficient ( -1 ) is 1 .The factors of the constant term (343) are 1 7 49 343 . Then the Rational Roots Tests yields the following possible solutions:
$$ \pm \frac{ 1 }{ 1 } , ~ \pm \frac{ 7 }{ 1 } , ~ \pm \frac{ 49 }{ 1 } , ~ \pm \frac{ 343 }{ 1 } ~ $$Substitute the POSSIBLE roots one by one into the polynomial to find the actual roots. Start first with the whole numbers.
If we plug these values into the polynomial $ P(x) $, we obtain $ P(7) = 0 $.
To find remaining zeros we use Factor Theorem. This theorem states that if $\frac{p}{q}$ is root of the polynomial then this polynomial can be divided with $ \color{blue}{q x - p} $. In this example:
Divide $ P(x) $ with $ \color{blue}{x - 7} $
$$ \frac{ -x^{3}+21x^{2}-147x+343 }{ \color{blue}{ x - 7 } } = -x^{2}+14x-49 $$Polynomial $ -x^{2}+14x-49 $ can be used to find the remaining roots.
$ \color{blue}{ -x^{2}+14x-49 } $ is a second degree polynomial. For a detailed answer how to find its roots you can use step-by-step quadratic equation solver.