Question

${a}^{4} - 3 {a}^{2} + 2 \: \: \: factorize$

Answer 1

HOLA USER ✌

HERE'S YOUR ANSWER FRIEND,

GIVEN QUESTION : a^4 - 3a² + 2 = 0

SOLUTION : a^4 - 2a² - a² + 2 = 0

==> a²(a² - 2) + (-1)(a² - 2)

THIS IMPLIES

➡ (a² - 1)

BY USING THE IDENTITY,

(a² - b²) = (a - b)(a + b)

IT CAN BE WRITTEN AS,

(a² - 1) =

(a² - 1²) = (a - 1)(a + 1)

AND,

➡ (a² - 2)

AGAIN WE CAN WRITE THIS

AS,

(a² - 2)(a - 1)(a + 1)

HENCE, FACTORISED.

⏫HOPE IT HELPS YOU.

Answer 2

Given equation : a^4 - 3a^2 + 2

Let a^2 = x and a^4 = x^2

Now, equation is x^2 - 3x + 2

⇒ x^2 - 3x + 2

⇒ x^2 - ( 2 + 1 )x + 2

⇒ x^2 - 2x - x + 2

⇒ x( x - 2 ) - ( x - 2 )

⇒ ( x - 2 )( x - 1 )

Substituting the value of x,

⇒ ( a^2 - 2 )( a^2 - 1 )

⇒ ( a^2 - ( √2 )^2  )( a^2 - 1^2 )

From factorization, we know that the value of a^2 - b^2 is ( a + b )( a - b ).

⇒ ( a - √2 )( a + √2 ) ( a + 1 )( a - 1 )

Or, ( a^2 - 2 )( a + 1 )( a - 1 )

Therefore,

a^4 - 3a^2 + 2 = ( a^2 - 2 )( a + 1 ) ( a - 1 ) or ( a - √2 )( a + √2 ) ( a + 1 )( a - 1 )