Calculating Function Values: Synthetic Division & Remainder Theorem

October 14, 2025 by TextBrain Team 68 views

Hey guys! Let's dive into a cool math topic: figuring out the values of a function using some neat tricks. Specifically, we're going to use synthetic division and the Remainder Theorem. These tools are super handy for evaluating polynomial functions without having to do a ton of direct substitution. We'll be looking at the function f(x) = 3x² - 7x + 7 and calculating its values at different points. Let's break it down step-by-step, so you can totally ace this!

Understanding the Remainder Theorem & Synthetic Division

Alright, before we start crunching numbers, let's get familiar with the players. The Remainder Theorem is like a shortcut. It tells us that if we divide a polynomial, f(x), by (x - c), the remainder is equal to f(c). Boom! That means to find the value of a function at a certain point (like f(2)), we can divide the polynomial by (x - 2), and the remainder will be our answer.

Now, how do we actually do the division? That's where synthetic division comes in. It's a simplified way of dividing polynomials, especially when dividing by a linear factor like (x - c). It's much faster and easier than long division. Instead of writing out all the xs and powers, we just work with the coefficients. Trust me, it's way less intimidating than it sounds. We'll see how to use it in a bit. So, in essence, synthetic division helps us apply the Remainder Theorem efficiently. The combination of these two methods makes finding function values a breeze.

Let's recap, the Remainder Theorem is the what and synthetic division is the how. It is important to understand both concepts before diving into this. This dynamic duo makes evaluating polynomials way more efficient and helps us avoid some tedious calculations. So, let's get started with our specific function, f(x) = 3x² - 7x + 7, and calculate f(-2), f(-1), f(0), f(1), and f(2) using these awesome techniques! You'll be a pro in no time, I swear!

Calculating f(-2) Using Synthetic Division and Remainder Theorem

Alright, let's start by calculating f(-2). According to the Remainder Theorem, we need to divide our function, f(x) = 3x² - 7x + 7, by (x - (-2)) or (x + 2), and the remainder will be our answer. Let's use synthetic division for this.

Here's how it works:

Set up the synthetic division: Write down the coefficients of our polynomial: 3, -7, and 7. Then, to the left, write the value of c, which is -2 (since we're dividing by (x + 2)). It should look like this:
```
-2 | 3  -7   7
```
Bring down the first coefficient: Bring down the first coefficient (3) below the line.
```
-2 | 3  -7   7
    ----- 
      3
```
Multiply and add: Multiply the number you just brought down (3) by -2 (our c value), and write the result (-6) under the next coefficient (-7).
```
-2 | 3  -7   7
      -6
    ----- 
      3
```
Add -7 and -6 to get -13. Write the result below the line.
```
-2 | 3  -7   7
      -6
    ----- 
      3 -13
```
Repeat: Multiply -13 by -2, which gives 26. Write 26 under the last coefficient (7).
```
-2 | 3  -7   7
      -6  26
    ----- 
      3 -13
```
Add 7 and 26 to get 33. Write the result below the line.
```
-2 | 3  -7   7
      -6  26
    ----- 
      3 -13  33
```

The last number in the bottom row (33) is our remainder. Therefore, according to the Remainder Theorem, f(-2) = 33. Easy peasy, right? We didn't have to plug -2 into the function directly, and we still found the answer! We can safely conclude that f(-2) = 33. It’s amazing how synthetic division streamlines the process, allowing us to efficiently evaluate polynomial functions. In essence, we have successfully used synthetic division and the Remainder Theorem to determine that when x = -2, the function f(x) equals 33.

Calculating f(-1) Using Synthetic Division and Remainder Theorem

Now, let's calculate f(-1). Again, we'll use synthetic division and the Remainder Theorem. This time, we're dividing f(x) = 3x² - 7x + 7 by (x - (-1)) or (x + 1).

Set up: Write down the coefficients: 3, -7, and 7. To the left, write -1 (because we're dividing by (x + 1)).
```
-1 | 3  -7   7
```
Bring down: Bring down the first coefficient (3).
```
-1 | 3  -7   7
    ----- 
      3
```
Multiply and add: Multiply 3 by -1, which gives -3. Write this under the -7.
```
-1 | 3  -7   7
      -3
    ----- 
      3
```
Add -7 and -3 to get -10. Write this below the line.
```
-1 | 3  -7   7
      -3
    ----- 
      3 -10
```
Repeat: Multiply -10 by -1, which gives 10. Write this under the 7.
```
-1 | 3  -7   7
      -3  10
    ----- 
      3 -10
```
Add 7 and 10 to get 17. Write this below the line.
```
-1 | 3  -7   7
      -3  10
    ----- 
      3 -10  17
```

The remainder is 17. Therefore, f(-1) = 17. Another one down! Synthetic division again made this super quick. Therefore, the result is f(-1) = 17. By utilizing synthetic division and the Remainder Theorem, we have efficiently determined the value of the function at x = -1.

Calculating f(0), f(1), and f(2)

Alright, we're on a roll! Let's finish up by calculating f(0), f(1), and f(2). We'll use the same synthetic division and Remainder Theorem approach.

Calculating f(0)

To find f(0), we divide f(x) = 3x² - 7x + 7 by (x - 0), or just x. Here’s the synthetic division:

Set up:
```
 0 | 3  -7   7
```
Bring down:
```
 0 | 3  -7   7
    ----- 
      3
```

Multiply and add:

 0 | 3  -7   7
      0
    ----- 
      3 -7

Repeat:

 0 | 3  -7   7
      0   0
    ----- 
      3 -7   7

The remainder is 7. Therefore, f(0) = 7. Notice that when x = 0, the function simply equals the constant term! Therefore, we can conclude that f(0) = 7.

Calculating f(1)

To find f(1), we divide f(x) = 3x² - 7x + 7 by (x - 1).

Set up:
```
 1 | 3  -7   7
```
Bring down:
```
 1 | 3  -7   7
    ----- 
      3
```

Multiply and add:

 1 | 3  -7   7
      3
    ----- 
      3 -4

Repeat:

 1 | 3  -7   7
      3  -4
    ----- 
      3 -4   3

The remainder is 3. Therefore, f(1) = 3. We're getting the hang of this! Therefore, the result is f(1) = 3.

Calculating f(2)

Finally, let's find f(2). We divide f(x) = 3x² - 7x + 7 by (x - 2).

Set up:
```
 2 | 3  -7   7
```
Bring down:
```
 2 | 3  -7   7
    ----- 
      3
```

Multiply and add:

 2 | 3  -7   7
      6
    ----- 
      3 -1

Repeat:

 2 | 3  -7   7
      6  -2
    ----- 
      3 -1   5

The remainder is 5. Therefore, f(2) = 5. And there you have it! We have successfully used the Remainder Theorem and synthetic division to evaluate the function at all the required points. Therefore, we can conclude that f(2) = 5.

Final Results

So, to recap, here are the values we calculated:

f(-2) = 33
f(-1) = 17
f(0) = 7
f(1) = 3
f(2) = 5

See? Synthetic division and the Remainder Theorem make these calculations a piece of cake! They're excellent tools to have in your math toolbox. Keep practicing, and you'll become a pro at evaluating polynomials in no time! Keep up the awesome work, guys!