Find roots of polynomial $$ p(x) = -8x^4+26x^3-8x^2-24x $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= 2\\[1 em]x_3 &= -\frac{ 3 }{ 4 }\\[1 em]x_4 &= 2 \end{aligned} $$Explanation
Step 1:
Factor out $ \color{blue}{ -2x }$ from $ -8x^4+26x^3-8x^2-24x $ and solve two separate equations:
$$ \begin{aligned} -8x^4+26x^3-8x^2-24x & = 0\\[1 em] \color{blue}{ -2x }\cdot ( 4x^3-13x^2+4x+12 ) & = 0 \\[1 em] \color{blue}{ -2x = 0} ~~ \text{or} ~~ 4x^3-13x^2+4x+12 & = 0 \end{aligned} $$One solution is $ \color{blue}{ x = 0 } $. Use second equation to find the remaining roots.
Step 2:
Use rational root test to find out that the $ \color{blue}{ x = 2 } $ is a root of polynomial $ 4x^3-13x^2+4x+12 $.
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}{ 12 } $, with factors of 1, 2, 3, 4, 6 and 12.
The leading coefficient is $ \color{red}{ 4 }$, with factors of 1, 2 and 4.
The POSSIBLE zeroes are:
$$ \begin{aligned} \dfrac{\color{blue}{p}}{\color{red}{q}} = & \dfrac{ \text{ factors of 12 }}{\text{ factors of 4 }} = \pm \dfrac{\text{ ( 1, 2, 3, 4, 6, 12 ) }}{\text{ ( 1, 2, 4 ) }} = \\[1 em] = & \pm \frac{ 1}{ 1} \pm \frac{ 2}{ 1} \pm \frac{ 3}{ 1} \pm \frac{ 4}{ 1} \pm \frac{ 6}{ 1} \pm \frac{ 12}{ 1} ~~ \pm \frac{ 1}{ 2} \pm \frac{ 2}{ 2} \pm \frac{ 3}{ 2} \pm \frac{ 4}{ 2} \pm \frac{ 6}{ 2} \pm \frac{ 12}{ 2} ~~ \pm \frac{ 1}{ 4} \pm \frac{ 2}{ 4} \pm \frac{ 3}{ 4} \pm \frac{ 4}{ 4} \pm \frac{ 6}{ 4} \pm \frac{ 12}{ 4} ~~ \end{aligned} $$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( 2 \right) = 0 $ so $ x = 2 $ 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-2 }$
$$ \frac{ 4x^3-13x^2+4x+12}{ x-2} = 4x^2-5x-6 $$Step 3:
The next rational root is $ x = 2 $
$$ \frac{ 4x^3-13x^2+4x+12}{ x-2} = 4x^2-5x-6 $$Step 4:
The next rational root is $ x = -\dfrac{ 3 }{ 4 } $
$$ \frac{ 4x^2-5x-6}{ 4x+3} = x-2 $$Step 5:
To find the last zero, solve equation $ x-2 = 0 $
$$ \begin{aligned} x-2 & = 0 \\[1 em] x & = 2 \end{aligned} $$