Learning A Math - remainder theorem

hello everyone, another series of understanding the "why" rather than just the "how". In this post I seek to explain how the Remainder Theorem works.

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The Remainder Theorem! It's a powerful tool in algebra.

What is the Remainder Theorem?

The Remainder Theorem states that when a polynomial f(x) is divided by (x - a), the remainder is equal to f(a).

Why does it work?

When doing long division, two important points to note:

1.      degree of quotient + degree of divisor = degree of dividend

2.      degree of remainder < degree of divisor

For any polynomial P(x) (dividend), when it is divided by f(x)  (divisor), We can write 

P(x) f(x)*Q(x) + R(x)

Dividend    (Divisor)(Quotient)  +  Remainder

where Q(x) is the quotient and R(x) the remainder. 

So when you are dividing by a linear divisor (degree 1), the remainder must be degree 0, a constant R

When you divide a polynomial f(x) by (x - a), you're essentially finding the quotient and remainder. The remainder is a constant value, let's call it R.

Since elementary years we already know the result from long division is written as:

Dividend = (Quotient) * (Divisor) + (Remainder)

Using polynomial long division or synthetic division, you can express f(x) as:

f(x) = (x - a)q(x) + R

where q(x) is the quotient.

Our goal is to solve for R, and in order to evaluate it despite not knowing what q(x) looks like we need a x-value that make than part of the expression 0.

Now, if you substitute x = a into this equation, you get:

f(a) = (a - a)q(a) + R

f(a) = 0 × q(a) + R

and hence R, the remainder is simply evaluated by f(a).

Extension

1) (x -a) was used to represent a linear divisor here, will anything just if we rewrite it as (ax - b)?

2) how will the theorem adjust if the divisor is a quadratic expression?