Question

show that the polynomial x⁴ 4x²+5 has no zero

​

Answer 1

$\sf\small\underline\red{Given:-}$

$f(x) = {x}^{4} + {4x}^{2} + 6$

$\sf\small\underline\red{To \: Find:-}$

Show that the polynomial has no zeros.

$\sf\small\underline\red{Solution:-}$

First we start making factors,

${x}^{4} + {4x}^{2} + 6 = 0$

${x}^{4} + {4x}^{2} + 4 + 2 = 0$

${x}^{4} + {4x}^{2} + {2}^{2} + 2 = 0$

Apply property,

$(a + b) ^{2} = {a}^{2} + {b}^{2} + 2ab( {x}^{2} + {2}^{2} ) ^{2} = 0$

Now,

$( {x}^{2} + {2}^{2} ) ^{2} = - 2$

Hence,There is no real solution.

Therefore, The given polynomial has no roots.