Question

Find roots of polynomial $p (x) = x^{3} - 48 x^{2} + 684 x - 2592$

Answer

The roots of polynomial $p (x)$ are:
$x_{1} x_{2} x_{3} = 6 = 18 = 24$

Explanation

Step 1:

Use rational root test to find out that the $x = 6$ is a root of polynomial $x^{3} - 48 x^{2} + 684 x - 2592$ .

The Rational Root Theorem tells us that if the polynomial has a rational zero then it must be a fraction $\frac{p}{q}$ , where $p$ is a factor of the constant term and $q$ is a factor of the leading coefficient.

The constant term is $2592$ , with a single factor of 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 72, 81, 96, 108, 144, 162, 216, 288, 324, 432, 648, 864, 1296 and 2592.

The leading coefficient is $1$ , with a single factor of 1.

The POSSIBLE zeroes are:

\frac{p}{q} = = \frac{factors of 2592}{factors of 1} = \pm \frac{( 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 72, 81, 96, 108, 144, 162, 216, 288, 324, 432, 648, 864, 1296, 2592 )}{( 1 )} = \pm \frac{1}{1} \pm \frac{2}{1} \pm \frac{3}{1} \pm \frac{4}{1} \pm \frac{6}{1} \pm \frac{8}{1} \pm \frac{9}{1} \pm \frac{12}{1} \pm \frac{16}{1} \pm \frac{18}{1} \pm \frac{24}{1} \pm \frac{27}{1} \pm \frac{32}{1} \pm \frac{36}{1} \pm \frac{48}{1} \pm \frac{54}{1} \pm \frac{72}{1} \pm \frac{81}{1} \pm \frac{96}{1} \pm \frac{108}{1} \pm \frac{144}{1} \pm \frac{162}{1} \pm \frac{216}{1} \pm \frac{288}{1} \pm \frac{324}{1} \pm \frac{432}{1} \pm \frac{648}{1} \pm \frac{864}{1} \pm \frac{1296}{1} \pm \frac{2592}{1}

Substitute the possible roots one by one into the polynomial to find the actual roots. Start first with the whole numbers.

We can see that $p (6) = 0$ so $x = 6$ is a root of a polynomial $p (x)$ .

To find remaining zeros we use Factor Theorem. This theorem states that if $\frac{p}{q}$ is root of the polynomial then the polynomial can be divided by $q x - p$ . In this example we divide polynomial $p$ by $x - 6$

\frac{x ^{3} - 48 x ^{2} + 684 x - 2592}{x - 6} = x^{2} - 42 x + 432

Step 2:

The next rational root is $x = 6$

\frac{x ^{3} - 48 x ^{2} + 684 x - 2592}{x - 6} = x^{2} - 42 x + 432

Step 3:

The next rational root is $x = 18$

\frac{x ^{2} - 42 x + 432}{x - 18} = x - 24

Step 4:

To find the last zero, solve equation $x - 24 = 0$

x - 24 x = 0 = 24

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Find roots of polynomial p(x)=x3−48x2+684x−2592

The roots of polynomial p(x) are: x1​x2​x3​​=6=18=24​

Find roots of polynomial $p (x) = x^{3} - 48 x^{2} + 684 x - 2592$

The roots of polynomial $p (x)$ are:
$x_{1} x_{2} x_{3} = 6 = 18 = 24$