Question

Factorize : x⁴+ x²+1​

Answer 1

Answer:

ANSWER

Given, x

4

+x

2

+1

=(x

2

)

2

+2(x

2

)(1)+1

2

−x

2

=(x

2

+1)

2

−x

2

=(x

2

+1−x)(x

2

+1+x)

Answer 2

Given:-

• $\sf x^4 + x^2 + 1$

We have to factorise this.

Answer:-

$\bf \large \: {x}^{4} + {x}^{2} + 1$

• We shall write it as,

$\longrightarrow \bf \large {( {x}^{2}) }^{2} + {x}^{2} + 1$

• Let $\sf y = x^2$

$\longrightarrow \bf \large {y}^{2} + y + 1$

We can write this as,

$\longrightarrow \bf \large {y}^{2} + 2y - y + 1$

$\longrightarrow \bf \large {y}^{2} + 2y + 1 - y$

• Notice that the first three terms are in the form of a² + 2ab + b² = (a + b)²

$\longrightarrow \bf \large (y + 1) {}^{2} - {( \sqrt{y})}^{2}$

• Using a² - b² = (a - b)(a + b)

$\longrightarrow \bf \large (y + 1 - \sqrt{y} )(y + 1 + \sqrt{y} )$

• But we had considered $\sf y = x^2$

$\bf \large \longrightarrow ( {x}^{2} + 1 - \sqrt{ {x}^{2} } )( {x}^{2} + 1 + \sqrt{ {x}^{2} } )$

$\bf \large \longrightarrow ( {x}^{2} + 1 - x )( {x}^{2} + 1 + x )$

$\longrightarrow \underline{ \underline{ \sf { \green{ ({x}^{2} - x + 1)( {x}^{2} + x + 1)}}}}$