Polynomial remainder theorem facts for kids

The Polynomial Remainder Theorem is a helpful rule in algebra. It tells us what the leftover part (called the remainder) will be when we divide a special math expression called a polynomial by a simple term like `x - a`.

Imagine you have a polynomial, which is like a math sentence with different powers of `x` (for example, `x^2 + 3x - 5`). If you divide this polynomial, let's call it `P(x)`, by `x - a` (where `a` is any regular number), the theorem says the remainder will be exactly `P(a)`. This means you just replace every `x` in the polynomial with the number `a` and calculate the result!

This theorem is quite easy to prove, and it's very useful for solving different math problems.

How it Works (The Proof)

Let's see why this theorem is true. When you divide a polynomial `P(x)` by `x - a`, you get a new polynomial, let's call it `Q(x)`, and a remainder, which we'll call `b`. We can write this like a division problem: `P(x) = Q(x) ⋅ (x - a) + b`

Now, let's imagine we replace `x` with the number `a` in this equation. `P(a) = Q(a) ⋅ (a - a) + b`

Since `a - a` is `0`, the equation becomes: `P(a) = Q(a) ⋅ 0 + b` `P(a) = 0 + b` `P(a) = b`

So, the remainder `b` is indeed equal to `P(a)`. This shows the theorem is correct!

A Simple Example

Let's try an example to see the Polynomial Remainder Theorem in action. Suppose we have the polynomial `P(x) = x^2 - 2x + 4`. We want to divide it by `x - 6`.

According to the theorem, the remainder should be `P(6)`. Let's calculate `P(6)`: `P(6) = (6)^2 - 2 ⋅ (6) + 4` `P(6) = 36 - 12 + 4` `P(6) = 28`

So, the theorem tells us the remainder will be `28`.

To check this, we can actually do the division. If `28` is the remainder, then `P(x) - 28` should divide perfectly by `x - 6`. `P(x) - 28 = (x^2 - 2x + 4) - 28` `P(x) - 28 = x^2 - 2x - 24`

Now, let's divide `x^2 - 2x - 24` by `x - 6`. We can factor `x^2 - 2x - 24` into `(x - 6)(x + 4)`. So, we have: `x^2 - 2x + 4 = (x + 4)(x - 6) + 28`

This confirms that when `x^2 - 2x + 4` is divided by `x - 6`, the remainder is `28`, just as the theorem predicted!

Why This Theorem is Useful

The Polynomial Remainder Theorem has several important uses in mathematics:

• Finding Remainders Quickly: Sometimes, a polynomial can be very long and complicated. Dividing it by `x - a` can take a lot of time. This theorem gives us a much faster way to find the remainder. We just need to plug in the value `a` into the polynomial.
• Theoretical Importance: This theorem is not just for quick calculations. It's a fundamental idea that helps prove other important theorems in algebra. For example, it's connected to the fundamental theorem of algebra, which is a big idea about the roots of polynomials. It also applies to more complex numbers, not just real numbers.

See also

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• In Spanish: Teorema del resto para niños