Find all zeros of the polynomial. P(x)=4x4+2x3−2x2−3x−1

Question

Find all zeros of the polynomial.  P(x)=4x4+2x3−2x2−3x−1

Donald Cheek · Accepted Answer

Step 1
Our Aim is to find all zeros of the polynomial given below:−

P (x) = 4 x^{4} + 2 x^{3} - 2 x^{2} - 3 x - 1

-(i)
Step 2
The left hand side of equation−(i) get factored into a product withthree terms:−

(x - 1) (2 x + 1) (2 x^{2} + 2 x + 1) = 0

Splitting the above terms into three equations, we have:−
x-1=0 or 2x+1-0 or

2 x^{2} + 2 x + 1 = 0

\Rightarrow x = 1

or

x = \frac{- 1}{2}

or

x^{2} + x + \frac{1}{2} = 0

\Rightarrow x = 1

or

x = \frac{- 1}{2}

or

x^{2} + x = \frac{- 1}{2}

\Rightarrow x = 1

or

x = \frac{- 1}{2}

or

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

\Rightarrow x = 1

or

x = \frac{- 1}{2}

or

{(x + \frac{1}{2})}^{2} = \frac{- 1}{4}

{(x + \frac{1}{2})}^{2} = \frac{- 1}{4}

has no solutions since for all x on the real line,

{(x + \frac{1}{2})}^{2} \geq 0

and

\frac{- 1}{4} < 0

.
Answer

\Rightarrow x = 1

or

x = \frac{- 1}{2}

Esther Phillips · Accepted Answer

Find all zeros of the polynomial. P(x)=4x^{4}+2x^{3}-2x^{2}-3x-1