Question

x⁴+x²+25
Factorise


Please​

Answer 1

Answer :-

The factors of the polynomial (x⁴ + x² + 25) are -

• (x² + 3x + 5) (x² - 3x + 5)

Solution :-

$\longrightarrow {x}^{4} + {x}^{2} + 25 \\ \\ \longrightarrow {x}^{4} + 10 {x}^{2} - 9 {x}^{2} + 25 \\ \\ \longrightarrow ({x}^{4} + 10 {x}^{2} + 25) - 9 {x}^{2} \\ \\ \longrightarrow \big( ( { {x}^{2} )}^{2} + 2 \times 5 \times {x}^{2} + ( {5})^{2} \big) - {(3x)}^{2}$

Using the identity :-

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

$\longrightarrow {( {x }^{2} + 5) }^{2} - ( {3x)}^{2}$

Now, Use the identity

• ${a}^{2} - {b}^{2} = (a + b)(a - b)$

$\longrightarrow \big( ({x}^{2} + 5) + 3x \big) \big( ({x}^{2} + 5) - 3x \big)$

Hence the Roots are ,

$</strong><strong>\longrightarrow</strong><strong> </strong><strong>( {x}^{2} + 3x + 5)( {x}^{2} - 3x + 5)$