When I first learned about the Remainder Theorem, I found it super helpful for figuring out polynomials. If you want to know how to use this theorem step-by-step, here’s what to do:

Step 1: Know Your Polynomial

First, figure out which polynomial you want to work with. For example, let’s use:

$P(x) = 2x^3 - 4x^2 + 3x - 5$

Step 2: Choose the Value of $c$

Next, decide what value of $c$ you will use to evaluate the polynomial. The Remainder Theorem says that if you want to find the remainder when dividing by $x - c$, just plug in that value of $c$. For instance, if you want to divide by $x - 2$, then $c = 2$.

Step 3: Plug $c$ into the Polynomial

Now, put $c$ into your polynomial. Using our example, replace $x$ in $P(x)$ with $2$:

$$P(2) = 2(2)^3 - 4(2)^2 + 3(2) - 5$$

Step 4: Do the Math

Next, calculate the value step-by-step. Here’s how:

1. First, do $2(2^3) = 2 \times 8 = 16$.
2. Then, calculate $-4(2^2) = -4 \times 4 = -16$.
3. Next, find $3(2) = 6$.
4. Now put it all together: $16 - 16 + 6 - 5$.

This simplifies to:

$$P(2) = 16 - 16 + 6 - 5 = 1$$

Step 5: Understand the Result

The final answer, which is $1$, is the remainder when you divide $P(x)$ by $x - 2$. If the remainder is zero, it means that $x - c$ is a factor of the polynomial!

Conclusion

And that’s all there is to it! By using the Remainder Theorem, you can quickly find remainders without doing long polynomial division. It makes things a lot easier. I hope this step-by-step guide helps you understand the theorem better!