Understand Polynomial Remainder when Divided by ( (x - a)^2 )

When dealing with polynomial division, one common question arises: if a polynomial ( p(x) ) leaves a remainder ( A ) when divided by ( x - a ), what is the remainder when the same polynomial is divided by ( x - a^2 )? This investigation will be detailed below, elaborating on the process step-by-step and providing a general solution.

Given Information and Initial Setup

Let's consider a polynomial ( p(x) ) such that when ( p(x) ) is divided by ( x - a ), the remainder is ( A ). This can be mathematically represented as:

[ p(x) (x - a)q(x) A ]

General Case Division

Now, we need to find the remainder when ( p(x) ) is divided by ( x - a^2 ). We can start from the initial expression and expand it, ensuring that we maintain the correct polynomial structure.

Equation Setup

We can rewrite ( p(x) ) as follows:

[ p(x) (x - a)^2 q(x) cdot k (x - a)A A ]

This can be further simplified as:

[ p(x) (x - a)^2 q(x) cdot k A cdot g(x) ]

where ( g(x) ) is a polynomial representing the remainder when divided by ( x - a^2 ).

Remainder Theorem Application

To solve for the remainder, we need to use the polynomial division technique. We can express ( p(x) ) as:

[ p(x) (x - a^2) d(x) (xa - a^2) q(x) cdot k (x - a)A A ]

Dividing Polynomials

Dividing ( p(x) ) by ( x - a^2 ), we can write:

[ frac{p(x)}{x - a^2} frac{(x - a)^2 q(x) cdot k A(x - a) A}{x - a^2} ]

This can be broken down further into:

[ frac{p(x)}{x - a^2} frac{(x - a)^2 q(x) cdot k}{x - a^2} frac{A(x - a) A}{x - a^2} ]

Which simplifies to:

[ frac{p(x)}{x - a^2} (x - a)q(x) cdot k A cdot frac{x - a 1}{x - a^2} ]

The term ( A cdot frac{x - a 1}{x - a^2} ) can be analyzed separately:

[ A cdot frac{x - a 1}{x - a^2} A cdot frac{1}{x - a} cdot frac{x - a 1}{x - a} ]

Solving for the remainder:

[ p(x) (x - a^2)left(q(x)k frac{A}{x - a}right) A ]

Final Answer

Therefore, the remainder when ( p(x) ) is divided by ( x - a^2 ) is simply ( A ).

So, the general formula for the remainder when ( p(x) ) is divided by ( x - a^2 ) is:

[ text{Remainder} A ]

This can be summarized as:

[ p(x) (x - a^2) q(x) A quad text{where the remainder is} quad A ]

Conclusion

In conclusion, the remainder when a polynomial ( p(x) ) is divided by ( x - a^2 ), given that the remainder is ( A ) when the same polynomial is divided by ( x - a ), is simply ( A ). This outcome is consistent with the remainder theorem and can be derived from the polynomial division process.