Find roots of polynomial $$ p(x) = -15x^8-60x^7+465x^6+2010x^5-3900x^4-21480x^3+480x^2+74880x+69120 $$

The roots of polynomial $ p(x) $ are:

$$ \begin{aligned}x_1 &= 3\\[1 em]x_2 &= 4\\[1 em]x_3 &= -2\\[1 em]x_4 &= -4\\[1 em]x_5 &= 3\\[1 em]x_6 &= -2\\[1 em]x_7 &= -4\\[1 em]x_8 &= -2 \end{aligned} $$

Step 1:

Use rational root test to find out that the $ \color{blue}{ x = 3 } $ is a root of polynomial $ -15x^8-60x^7+465x^6+2010x^5-3900x^4-21480x^3+480x^2+74880x+69120 $.

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

The constant term is $ \color{blue}{ 69120 } $, with factors of 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 27, 30, 32, 36, 40, 45, 48, 54, 60, 64, 72, 80, 90, 96, 108, 120, 128, 135, 144, 160, 180, 192, 216, 240, 256, 270, 288, 320, 360, 384, 432, 480, 512, 540, 576, 640, 720, 768, 864, 960, 1080, 1152, 1280, 1440, 1536, 1728, 1920, 2160, 2304, 2560, 2880, 3456, 3840, 4320, 4608, 5760, 6912, 7680, 8640, 11520, 13824, 17280, 23040, 34560 and 69120.

The leading coefficient is $ \color{red}{ 15 }$, with factors of 1, 3, 5 and 15.

The POSSIBLE zeroes are:

$$ \begin{aligned} \dfrac{\color{blue}{p}}{\color{red}{q}} = & \dfrac{ \text{ factors of 69120 }}{\text{ factors of 15 }} = \pm \dfrac{\text{ ( 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 27, 30, 32, 36, 40, 45, 48, 54, 60, 64, 72, 80, 90, 96, 108, 120, 128, 135, 144, 160, 180, 192, 216, 240, 256, 270, 288, 320, 360, 384, 432, 480, 512, 540, 576, 640, 720, 768, 864, 960, 1080, 1152, 1280, 1440, 1536, 1728, 1920, 2160, 2304, 2560, 2880, 3456, 3840, 4320, 4608, 5760, 6912, 7680, 8640, 11520, 13824, 17280, 23040, 34560, 69120 ) }}{\text{ ( 1, 3, 5, 15 ) }} = \\[1 em] = & \pm \frac{ 1}{ 1} \pm \frac{ 2}{ 1} \pm \frac{ 3}{ 1} \pm \frac{ 4}{ 1} \pm \frac{ 5}{ 1} \pm \frac{ 6}{ 1} \pm \frac{ 8}{ 1} \pm \frac{ 9}{ 1} \pm \frac{ 10}{ 1} \pm \frac{ 12}{ 1} \pm \frac{ 15}{ 1} \pm \frac{ 16}{ 1} \pm \frac{ 18}{ 1} \pm \frac{ 20}{ 1} \pm \frac{ 24}{ 1} \pm \frac{ 27}{ 1} \pm \frac{ 30}{ 1} \pm \frac{ 32}{ 1} \pm \frac{ 36}{ 1} \pm \frac{ 40}{ 1} \pm \frac{ 45}{ 1} \pm \frac{ 48}{ 1} \pm \frac{ 54}{ 1} \pm \frac{ 60}{ 1} \pm \frac{ 64}{ 1} \pm \frac{ 72}{ 1} \pm \frac{ 80}{ 1} \pm \frac{ 90}{ 1} \pm \frac{ 96}{ 1} \pm \frac{ 108}{ 1} \pm \frac{ 120}{ 1} \pm \frac{ 128}{ 1} \pm \frac{ 135}{ 1} \pm \frac{ 144}{ 1} \pm \frac{ 160}{ 1} \pm \frac{ 180}{ 1} \pm \frac{ 192}{ 1} \pm \frac{ 216}{ 1} \pm \frac{ 240}{ 1} \pm \frac{ 256}{ 1} \pm \frac{ 270}{ 1} \pm \frac{ 288}{ 1} \pm \frac{ 320}{ 1} \pm \frac{ 360}{ 1} \pm \frac{ 384}{ 1} \pm \frac{ 432}{ 1} \pm \frac{ 480}{ 1} \pm \frac{ 512}{ 1} \pm \frac{ 540}{ 1} \pm \frac{ 576}{ 1} \pm \frac{ 640}{ 1} \pm \frac{ 720}{ 1} \pm \frac{ 768}{ 1} \pm \frac{ 864}{ 1} \pm \frac{ 960}{ 1} \pm \frac{ 1080}{ 1} \pm \frac{ 1152}{ 1} \pm \frac{ 1280}{ 1} \pm \frac{ 1440}{ 1} \pm \frac{ 1536}{ 1} \pm \frac{ 1728}{ 1} \pm \frac{ 1920}{ 1} \pm \frac{ 2160}{ 1} \pm \frac{ 2304}{ 1} \pm \frac{ 2560}{ 1} \pm \frac{ 2880}{ 1} \pm \frac{ 3456}{ 1} \pm \frac{ 3840}{ 1} \pm \frac{ 4320}{ 1} \pm \frac{ 4608}{ 1} \pm \frac{ 5760}{ 1} \pm \frac{ 6912}{ 1} \pm \frac{ 7680}{ 1} \pm \frac{ 8640}{ 1} \pm \frac{ 11520}{ 1} \pm \frac{ 13824}{ 1} \pm \frac{ 17280}{ 1} \pm \frac{ 23040}{ 1} \pm \frac{ 34560}{ 1} \pm \frac{ 69120}{ 1} ~~ \pm \frac{ 1}{ 3} \pm \frac{ 2}{ 3} \pm \frac{ 3}{ 3} \pm \frac{ 4}{ 3} \pm \frac{ 5}{ 3} \pm \frac{ 6}{ 3} \pm \frac{ 8}{ 3} \pm \frac{ 9}{ 3} \pm \frac{ 10}{ 3} \pm \frac{ 12}{ 3} \pm \frac{ 15}{ 3} \pm \frac{ 16}{ 3} \pm \frac{ 18}{ 3} \pm \frac{ 20}{ 3} \pm \frac{ 24}{ 3} \pm \frac{ 27}{ 3} \pm \frac{ 30}{ 3} \pm \frac{ 32}{ 3} \pm \frac{ 36}{ 3} \pm \frac{ 40}{ 3} \pm \frac{ 45}{ 3} \pm \frac{ 48}{ 3} \pm \frac{ 54}{ 3} \pm \frac{ 60}{ 3} \pm \frac{ 64}{ 3} \pm \frac{ 72}{ 3} \pm \frac{ 80}{ 3} \pm \frac{ 90}{ 3} \pm \frac{ 96}{ 3} \pm \frac{ 108}{ 3} \pm \frac{ 120}{ 3} \pm \frac{ 128}{ 3} \pm \frac{ 135}{ 3} \pm \frac{ 144}{ 3} \pm \frac{ 160}{ 3} \pm \frac{ 180}{ 3} \pm \frac{ 192}{ 3} \pm \frac{ 216}{ 3} \pm \frac{ 240}{ 3} \pm \frac{ 256}{ 3} \pm \frac{ 270}{ 3} \pm \frac{ 288}{ 3} \pm \frac{ 320}{ 3} \pm \frac{ 360}{ 3} \pm \frac{ 384}{ 3} \pm \frac{ 432}{ 3} \pm \frac{ 480}{ 3} \pm \frac{ 512}{ 3} \pm \frac{ 540}{ 3} \pm \frac{ 576}{ 3} \pm \frac{ 640}{ 3} \pm \frac{ 720}{ 3} \pm \frac{ 768}{ 3} \pm \frac{ 864}{ 3} \pm \frac{ 960}{ 3} \pm \frac{ 1080}{ 3} \pm \frac{ 1152}{ 3} \pm \frac{ 1280}{ 3} \pm \frac{ 1440}{ 3} \pm \frac{ 1536}{ 3} \pm \frac{ 1728}{ 3} \pm \frac{ 1920}{ 3} \pm \frac{ 2160}{ 3} \pm \frac{ 2304}{ 3} \pm \frac{ 2560}{ 3} \pm \frac{ 2880}{ 3} \pm \frac{ 3456}{ 3} \pm \frac{ 3840}{ 3} \pm \frac{ 4320}{ 3} \pm \frac{ 4608}{ 3} \pm \frac{ 5760}{ 3} \pm \frac{ 6912}{ 3} \pm \frac{ 7680}{ 3} \pm \frac{ 8640}{ 3} \pm \frac{ 11520}{ 3} \pm \frac{ 13824}{ 3} \pm \frac{ 17280}{ 3} \pm \frac{ 23040}{ 3} \pm \frac{ 34560}{ 3} \pm \frac{ 69120}{ 3} ~~ \pm \frac{ 1}{ 5} \pm \frac{ 2}{ 5} \pm \frac{ 3}{ 5} \pm \frac{ 4}{ 5} \pm \frac{ 5}{ 5} \pm \frac{ 6}{ 5} \pm \frac{ 8}{ 5} \pm \frac{ 9}{ 5} \pm \frac{ 10}{ 5} \pm \frac{ 12}{ 5} \pm \frac{ 15}{ 5} \pm \frac{ 16}{ 5} \pm \frac{ 18}{ 5} \pm \frac{ 20}{ 5} \pm \frac{ 24}{ 5} \pm \frac{ 27}{ 5} \pm \frac{ 30}{ 5} \pm \frac{ 32}{ 5} \pm \frac{ 36}{ 5} \pm \frac{ 40}{ 5} \pm \frac{ 45}{ 5} \pm \frac{ 48}{ 5} \pm \frac{ 54}{ 5} \pm \frac{ 60}{ 5} \pm \frac{ 64}{ 5} \pm \frac{ 72}{ 5} \pm \frac{ 80}{ 5} \pm \frac{ 90}{ 5} \pm \frac{ 96}{ 5} \pm \frac{ 108}{ 5} \pm \frac{ 120}{ 5} \pm \frac{ 128}{ 5} \pm \frac{ 135}{ 5} \pm \frac{ 144}{ 5} \pm \frac{ 160}{ 5} \pm \frac{ 180}{ 5} \pm \frac{ 192}{ 5} \pm \frac{ 216}{ 5} \pm \frac{ 240}{ 5} \pm \frac{ 256}{ 5} \pm \frac{ 270}{ 5} \pm \frac{ 288}{ 5} \pm \frac{ 320}{ 5} \pm \frac{ 360}{ 5} \pm \frac{ 384}{ 5} \pm \frac{ 432}{ 5} \pm \frac{ 480}{ 5} \pm \frac{ 512}{ 5} \pm \frac{ 540}{ 5} \pm \frac{ 576}{ 5} \pm \frac{ 640}{ 5} \pm \frac{ 720}{ 5} \pm \frac{ 768}{ 5} \pm \frac{ 864}{ 5} \pm \frac{ 960}{ 5} \pm \frac{ 1080}{ 5} \pm \frac{ 1152}{ 5} \pm \frac{ 1280}{ 5} \pm \frac{ 1440}{ 5} \pm \frac{ 1536}{ 5} \pm \frac{ 1728}{ 5} \pm \frac{ 1920}{ 5} \pm \frac{ 2160}{ 5} \pm \frac{ 2304}{ 5} \pm \frac{ 2560}{ 5} \pm \frac{ 2880}{ 5} \pm \frac{ 3456}{ 5} \pm \frac{ 3840}{ 5} \pm \frac{ 4320}{ 5} \pm \frac{ 4608}{ 5} \pm \frac{ 5760}{ 5} \pm \frac{ 6912}{ 5} \pm \frac{ 7680}{ 5} \pm \frac{ 8640}{ 5} \pm \frac{ 11520}{ 5} \pm \frac{ 13824}{ 5} \pm \frac{ 17280}{ 5} \pm \frac{ 23040}{ 5} \pm \frac{ 34560}{ 5} \pm \frac{ 69120}{ 5} ~~ \pm \frac{ 1}{ 15} \pm \frac{ 2}{ 15} \pm \frac{ 3}{ 15} \pm \frac{ 4}{ 15} \pm \frac{ 5}{ 15} \pm \frac{ 6}{ 15} \pm \frac{ 8}{ 15} \pm \frac{ 9}{ 15} \pm \frac{ 10}{ 15} \pm \frac{ 12}{ 15} \pm \frac{ 15}{ 15} \pm \frac{ 16}{ 15} \pm \frac{ 18}{ 15} \pm \frac{ 20}{ 15} \pm \frac{ 24}{ 15} \pm \frac{ 27}{ 15} \pm \frac{ 30}{ 15} \pm \frac{ 32}{ 15} \pm \frac{ 36}{ 15} \pm \frac{ 40}{ 15} \pm \frac{ 45}{ 15} \pm \frac{ 48}{ 15} \pm \frac{ 54}{ 15} \pm \frac{ 60}{ 15} \pm \frac{ 64}{ 15} \pm \frac{ 72}{ 15} \pm \frac{ 80}{ 15} \pm \frac{ 90}{ 15} \pm \frac{ 96}{ 15} \pm \frac{ 108}{ 15} \pm \frac{ 120}{ 15} \pm \frac{ 128}{ 15} \pm \frac{ 135}{ 15} \pm \frac{ 144}{ 15} \pm \frac{ 160}{ 15} \pm \frac{ 180}{ 15} \pm \frac{ 192}{ 15} \pm \frac{ 216}{ 15} \pm \frac{ 240}{ 15} \pm \frac{ 256}{ 15} \pm \frac{ 270}{ 15} \pm \frac{ 288}{ 15} \pm \frac{ 320}{ 15} \pm \frac{ 360}{ 15} \pm \frac{ 384}{ 15} \pm \frac{ 432}{ 15} \pm \frac{ 480}{ 15} \pm \frac{ 512}{ 15} \pm \frac{ 540}{ 15} \pm \frac{ 576}{ 15} \pm \frac{ 640}{ 15} \pm \frac{ 720}{ 15} \pm \frac{ 768}{ 15} \pm \frac{ 864}{ 15} \pm \frac{ 960}{ 15} \pm \frac{ 1080}{ 15} \pm \frac{ 1152}{ 15} \pm \frac{ 1280}{ 15} \pm \frac{ 1440}{ 15} \pm \frac{ 1536}{ 15} \pm \frac{ 1728}{ 15} \pm \frac{ 1920}{ 15} \pm \frac{ 2160}{ 15} \pm \frac{ 2304}{ 15} \pm \frac{ 2560}{ 15} \pm \frac{ 2880}{ 15} \pm \frac{ 3456}{ 15} \pm \frac{ 3840}{ 15} \pm \frac{ 4320}{ 15} \pm \frac{ 4608}{ 15} \pm \frac{ 5760}{ 15} \pm \frac{ 6912}{ 15} \pm \frac{ 7680}{ 15} \pm \frac{ 8640}{ 15} \pm \frac{ 11520}{ 15} \pm \frac{ 13824}{ 15} \pm \frac{ 17280}{ 15} \pm \frac{ 23040}{ 15} \pm \frac{ 34560}{ 15} \pm \frac{ 69120}{ 15} ~~ \end{aligned} $$

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\left( 3 \right) = 0 $ so $ x = 3 $ is a root of a polynomial $ p(x) $.

To find remaining zeros we use Factor Theorem. This theorem states that if $ \dfrac{p}{q} $ is root of the polynomial then the polynomial can be divided by $ \color{blue}{qx − p} $. In this example we divide polynomial $ p $ by $ \color{blue}{ x-3 }$

$$ \frac{ -15x^8-60x^7+465x^6+2010x^5-3900x^4-21480x^3+480x^2+74880x+69120}{ x-3} = -15x^7-105x^6+150x^5+2460x^4+3480x^3-11040x^2-32640x-23040 $$

Step 2:

The next rational root is $ x = 3 $

$$ \frac{ -15x^8-60x^7+465x^6+2010x^5-3900x^4-21480x^3+480x^2+74880x+69120}{ x-3} = -15x^7-105x^6+150x^5+2460x^4+3480x^3-11040x^2-32640x-23040 $$

Step 3:

The next rational root is $ x = 4 $

$$ \frac{ -15x^7-105x^6+150x^5+2460x^4+3480x^3-11040x^2-32640x-23040}{ x-4} = -15x^6-165x^5-510x^4+420x^3+5160x^2+9600x+5760 $$

Step 4:

The next rational root is $ x = -2 $

$$ \frac{ -15x^6-165x^5-510x^4+420x^3+5160x^2+9600x+5760}{ x+2} = -15x^5-135x^4-240x^3+900x^2+3360x+2880 $$

Step 5:

The next rational root is $ x = -4 $

$$ \frac{ -15x^5-135x^4-240x^3+900x^2+3360x+2880}{ x+4} = -15x^4-75x^3+60x^2+660x+720 $$

Step 6:

The next rational root is $ x = 3 $

$$ \frac{ -15x^4-75x^3+60x^2+660x+720}{ x-3} = -15x^3-120x^2-300x-240 $$

Step 7:

The next rational root is $ x = -2 $

$$ \frac{ -15x^3-120x^2-300x-240}{ x+2} = -15x^2-90x-120 $$

Step 8:

The next rational root is $ x = -4 $

$$ \frac{ -15x^2-90x-120}{ x+4} = -15x-30 $$

Step 9:

To find the last zero, solve equation $ -15x-30 = 0 $

$$ \begin{aligned} -15x-30 & = 0 \\[1 em] -15 \cdot x & = 30 \\[1 em] x & = \frac{ 30 }{ -15 } \\[1 em] x & = \frac{ 30 : 15 }{ -15 : 15} \\[1 em] x & = -2 \end{aligned} $$

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