Find roots of polynomial $$ p(x) = 10x^4-57x^3+51x^2+27x-7 $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= -\frac{ 1 }{ 2 }\\[1 em]x_2 &= \frac{ 1 }{ 5 }\\[1 em]x_3 &= 4.4142\\[1 em]x_4 &= 1.5858 \end{aligned} $$Explanation
Step 1:
Use rational root test to find out that the $ \color{blue}{ x = -\dfrac{ 1 }{ 2 } } $ is a root of polynomial $ 10x^4-57x^3+51x^2+27x-7 $.
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}{ 7 } $, with factors of 1 and 7.
The leading coefficient is $ \color{red}{ 10 }$, with factors of 1, 2, 5 and 10.
The POSSIBLE zeroes are:
$$ \begin{aligned} \dfrac{\color{blue}{p}}{\color{red}{q}} = & \dfrac{ \text{ factors of 7 }}{\text{ factors of 10 }} = \pm \dfrac{\text{ ( 1, 7 ) }}{\text{ ( 1, 2, 5, 10 ) }} = \\[1 em] = & \pm \frac{ 1}{ 1} \pm \frac{ 7}{ 1} ~~ \pm \frac{ 1}{ 2} \pm \frac{ 7}{ 2} ~~ \pm \frac{ 1}{ 5} \pm \frac{ 7}{ 5} ~~ \pm \frac{ 1}{ 10} \pm \frac{ 7}{ 10}\\[ 1 em] \end{aligned} $$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( -\dfrac{ 1 }{ 2 } \right) = 0 $ so $ x = -\dfrac{ 1 }{ 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}{ 2x+1 }$
$$ \frac{ 10x^4-57x^3+51x^2+27x-7}{ 2x+1} = 5x^3-31x^2+41x-7 $$Step 2:
The next rational root is $ x = -\dfrac{ 1 }{ 2 } $
$$ \frac{ 10x^4-57x^3+51x^2+27x-7}{ 2x+1} = 5x^3-31x^2+41x-7 $$Step 3:
The next rational root is $ x = \dfrac{ 1 }{ 5 } $
$$ \frac{ 5x^3-31x^2+41x-7}{ 5x-1} = x^2-6x+7 $$Step 4:
The solutions of $ x^2-6x+7 = 0 $ are: $ x = 3-\sqrt{ 2 } ~ \text{and} ~ x = 3+\sqrt{ 2 }$.
You can use step-by-step quadratic equation solver to see a detailed explanation on how to solve this quadratic equation.