Find roots of polynomial $$ p(x) = 16x^4-41x^2+25 $$

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

$$ \begin{aligned}x_1 &= 1\\[1 em]x_2 &= -1\\[1 em]x_3 &= \frac{ 5 }{ 4 }\\[1 em]x_4 &= -\frac{ 5 }{ 4 } \end{aligned} $$

Step 1:

Use rational root test to find out that the $ \color{blue}{ x = 1 } $ is a root of polynomial $ 16x^4-41x^2+25 $.

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}{ 25 } $, with factors of 1, 5 and 25.

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

The POSSIBLE zeroes are:

$$ \begin{aligned} \dfrac{\color{blue}{p}}{\color{red}{q}} = & \dfrac{ \text{ factors of 25 }}{\text{ factors of 16 }} = \pm \dfrac{\text{ ( 1, 5, 25 ) }}{\text{ ( 1, 2, 4, 8, 16 ) }} = \\[1 em] = & \pm \frac{ 1}{ 1} \pm \frac{ 5}{ 1} \pm \frac{ 25}{ 1} ~~ \pm \frac{ 1}{ 2} \pm \frac{ 5}{ 2} \pm \frac{ 25}{ 2} ~~ \pm \frac{ 1}{ 4} \pm \frac{ 5}{ 4} \pm \frac{ 25}{ 4}\\[ 1 em] \pm \frac{ 1}{ 8} & \pm \frac{ 5}{ 8} & \pm \frac{ 25}{ 8} ~~ \pm \frac{ 1}{ 16} \pm \frac{ 5}{ 16} \pm \frac{ 25}{ 16} ~~ \end{aligned} $$

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( 1 \right) = 0 $ so $ x = 1 $ 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-1 }$

$$ \frac{ 16x^4-41x^2+25}{ x-1} = 16x^3+16x^2-25x-25 $$

Step 2:

The next rational root is $ x = 1 $

$$ \frac{ 16x^4-41x^2+25}{ x-1} = 16x^3+16x^2-25x-25 $$

Step 3:

The next rational root is $ x = -1 $

$$ \frac{ 16x^3+16x^2-25x-25}{ x+1} = 16x^2-25 $$

Step 4:

The next rational root is $ x = \dfrac{ 5 }{ 4 } $

$$ \frac{ 16x^2-25}{ 4x-5} = 4x+5 $$

Step 5:

To find the last zero, solve equation $ 4x+5 = 0 $

$$ \begin{aligned} 4x+5 & = 0 \\[1 em] 4 \cdot x & = -5 \\[1 em] x & = - \frac{ 5 }{ 4 } \end{aligned} $$

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