Question

Find the factors of $$x^{4}+9x^{2}+81$$.

Answer 1

Hint: In this question it is given that we have to find the factors of $$x^{4}+9x^{2}+81$$. So to find the solution we need to express the above polynomial into the multiplication of algebraic expression, so for this we have to observe whether the algebraic expression is following any identity or not,
Complete step-by-step solution:
Given expression,
$$x^{4}+9x^{2}+81$$
This can be expressed as,
$$\left( x^{2}\right)^{2} +9\times x^{2}+9^{2}$$ [$$\because a^{nm}=\left( a^{n}\right)^{m} $$]
=$$\left( x^{2}\right)^{2} +2\times 9x^{2}+9^{2}-9x^{2}$$..........………………(1). [we just added and subtracted $$9x^{2}$$]
Now as we know the identity, $$a^{2}+2ab+b^{2}=\left( a+b\right)^{2} $$
So if we take $$a=x^{2},b=9$$ then by above formula we can write,
$$\left( x^{2}\right)^{2} +2\times x^{2}\times 9+9^{2}=\left( x^{2}+9\right)^{2} $$........(2)
From (1) we have,
$$\{ \left( x^{2}\right)^{2} +2\times x^{2}\times 9+9^{2}\} -9x^{2}$$
=$$\left( x^{2}+9\right)^{2} -9x^{2}$$
=$$\left( x^{2}+9\right)^{2} -3^{2}\times x^{2}$$
=$$\left( x^{2}+9\right)^{2} -\left( 3x\right)^{2} $$...................................(2). [$$\because a^{n}\times b^{n}=\left( ab\right)^{n} $$]

Now we are going to use another identity, i.e, $$a^{2}-b^{2}=\left( a+b\right) \left( a-b\right) $$
So by this identity, (2) can be written as,
$$\left( x^{2}+9\right)^{2} -\left( 3x\right)^{2} $$
=$$\left( x^{2}+9+3x\right) \left( x^{2}+9-3x\right) $$
So therefore we get,
$$x^{4}+9x^{2}+81$$=$$\left( x^{2}+9+3x\right) \left( x^{2}+9-3x\right) $$.
Thus the factors are $$\left( x^{2}+9+3x\right) \ and\ \left( x^{2}+9-3x\right) $$.
Note: So to find the solution you need to have the basic idea about factors which states that if algebraic expressions are expressed as the product of numbers, algebraic variables or algebraic expressions, then each of these numbers and expressions is called the factor of algebraic expressions. So because of this we have used these identities,
i.e, $$a^{2}+2ab+b^{2}=\left( a+b\right)^{2} =\left( a+b\right) \left( a+b\right) $$
$$a^{2}-b^{2}=\left( a+b\right) \left( a-b\right) $$
So by this we can express algebraic expression as a product of two algebraic expressions which we called as factors.