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b) Factorize completely a⁴ _ 8a-9.​

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Factorise: a4−b4

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Hint: In this type of question, factorise the expression by making terms of the expression perfect square and then by using the identity of x2−y2 . This identity is applicable for the equation having subtraction between the values which are perfect squares.

Complete step-by-step answer:

The given expression for factorisation is a4−b4 .

Here we see that a4 and b4 have the exponent of 4. To make the expression as quadratic as well make the terms perfect square we can write the given expression in this way:

(a2)2−(b2)2

The given expression has the degree of 2, as the highest exponent in the expression is 2. Now we check the end terms of the expression which are (a2)2 and (b2)2 .

Here, we found that (a2)2 is the perfect square of a2

but (b2)2 is a perfect square of b2 , hence by using the identity of x2−y2 to find the factor of the given expression.

So, as we know that,

x2−y2=(x+y)(x−y)

Comparing the given expression with the identity then we get,

x=a2,y=b2

Substituting the values of x and y in the identity then we get,

⇒(a2)2−(b2)2=(a2+b2)(a2−b2)⇒a4−b4=(a2+b2)(a2−b2)

Now, again apply the identity x2−y2 for (a2−b2) . On comparing the expression with identity then we get,

x=a,y=b

After substituting the values we get,

a2−b2=(a+b)(a−b)

Substituting the value of (a2−b2) in the above expression then we get,

⇒a4−b4=(a2+b2)(a+b)(a−b)

Therefore, (a2+b2)(a+b)(a−b) is the required factor of the given expression a4−b4 and a4−b4 completely divisible by these factors.