Find roots of polynomial $$ p(x) = 64x^3-48x^2+12x-1 $$

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

$$ \begin{aligned}x_1 &= \frac{ 1 }{ 4 }\\[1 em]x_2 &= \frac{ 1 }{ 4 }\\[1 em]x_3 &= \frac{ 1 }{ 4 } \end{aligned} $$

Step 1:

Use rational root test to find out that the $ \color{blue}{ x = \dfrac{ 1 }{ 4 } } $ is a root of polynomial $ 64x^3-48x^2+12x-1 $.

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}{ 1 } $, with factors of 1.

The leading coefficient is $ \color{red}{ 64 }$, with factors of 1, 2, 4, 8, 16, 32 and 64.

The POSSIBLE zeroes are:

$$ \begin{aligned} \dfrac{\color{blue}{p}}{\color{red}{q}} = & \dfrac{ \text{ factors of 1 }}{\text{ factors of 64 }} = \pm \dfrac{\text{ ( 1 ) }}{\text{ ( 1, 2, 4, 8, 16, 32, 64 ) }} = \\[1 em] = & \pm \frac{ 1}{ 1} ~~ \pm \frac{ 1}{ 2} ~~ \pm \frac{ 1}{ 4} ~~ \pm \frac{ 1}{ 8} ~~ \pm \frac{ 1}{ 16} ~~ \pm \frac{ 1}{ 32} ~~ \pm \frac{ 1}{ 64} ~~ \end{aligned} $$

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 }{ 4 } \right) = 0 $ so $ x = \dfrac{ 1 }{ 4 } $ 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}{ 4x-1 }$

$$ \frac{ 64x^3-48x^2+12x-1}{ 4x-1} = 16x^2-8x+1 $$

Step 2:

The next rational root is $ x = \dfrac{ 1 }{ 4 } $

$$ \frac{ 64x^3-48x^2+12x-1}{ 4x-1} = 16x^2-8x+1 $$

Step 3:

The next rational root is $ x = \dfrac{ 1 }{ 4 } $

$$ \frac{ 16x^2-8x+1}{ 4x-1} = 4x-1 $$

Step 4:

To find the last zero, solve equation $ 4x-1 = 0 $

$$ \begin{aligned} 4x-1 & = 0 \\[1 em] 4 \cdot x & = 1 \\[1 em] x & = \frac{ 1 }{ 4 } \end{aligned} $$

Polynomial Roots Calculator