Synthetic Division: Finding Zeros Of Polynomial Functions

Oct 18, 2025 by ADMIN 58 views

Hey everyone! Today, we're diving into a cool math trick called synthetic division. We'll use it to figure out if some specific numbers are the zeros of a polynomial function. Basically, we're trying to find the values of x that make the function equal to zero. Let's get started with our example: $f(x) = 3x^3 + 7x^2 - 14x + 24$ , and we need to check if -4 and 3 are zeros. Synthetic division is like a shortcut for polynomial division, making things a lot easier. It's particularly useful when you're checking potential zeros because it quickly tells you the remainder. If the remainder is zero, the number you're testing is a zero of the function. Ready to get started? Let’s break it down step-by-step.

Understanding Zeros of Polynomial Functions

Before we jump into synthetic division, let's make sure we're all on the same page about what zeros are. A zero of a polynomial function is simply a value of x that makes the function equal to zero. Think of it as the x-intercept of the graph of the function – the point where the graph crosses the x-axis. Finding zeros is super important because it helps us understand the behavior of the polynomial and solve related equations.

So, why do we care about zeros? Well, they give us a lot of information. They tell us where the function crosses the x-axis, which helps in graphing. They also help us factor the polynomial. Knowing the zeros, we can express the polynomial as a product of linear factors. For instance, if we find that x = -4 is a zero, then $(x + 4)$ is a factor of the polynomial. This factoring can be really useful for solving polynomial equations and simplifying expressions.

Moreover, the zeros of a function are crucial in real-world applications. Imagine a projectile motion problem or a business trying to model its profit. Polynomial functions are often used in these scenarios. Knowing the zeros can help determine key points such as when a projectile hits the ground or when a company breaks even. Thus, understanding and finding zeros is a fundamental skill in algebra and calculus, opening doors to advanced mathematical concepts and practical problem-solving.

For our example, $f(x) = 3x^3 + 7x^2 - 14x + 24$ , we want to figure out if -4 and 3 are zeros. This means we're checking if $f(-4) = 0$ and $f(3) = 0$ . We could plug in those values directly into the equation. But synthetic division is a much cleaner and quicker way to do it, and it gives us extra information about the polynomial in the process. We will see, when we are done, if the remainder is 0. If it is, then the value we tested is a zero. If the remainder is not 0, then the value we tested is not a zero.

Applying Synthetic Division to Find Zeros

Alright, let's get our hands dirty with synthetic division! First, let's test if -4 is a zero. Here’s how it goes:

Set up the problem. Write down the coefficients of the polynomial. In our case, the coefficients are 3, 7, -14, and 24. Write down the value we are testing, -4, to the left. It should look like this:

-4 | 3   7  -14  24
    |_________________

Bring down the first coefficient. Bring down the first coefficient (3) below the line.

-4 | 3   7  -14  24
    |_________________
     3

Multiply and add. Multiply the number we brought down (3) by the value we're testing (-4), which gives us -12. Write -12 under the next coefficient (7) and add them together (7 + (-12) = -5).

-4 | 3   7  -14  24
    |    -12
    |_________________
     3  -5

Repeat. Multiply -5 by -4, which equals 20. Write 20 under -14 and add them together (-14 + 20 = 6).

-4 | 3   7  -14  24
    |    -12   20
    |_________________
     3  -5   6

Repeat again. Multiply 6 by -4, which equals -24. Write -24 under 24 and add them together (24 + (-24) = 0).

-4 | 3   7  -14  24
    |    -12   20  -24
    |_________________
     3  -5   6   0

The last number in the bottom row is the remainder. In this case, the remainder is 0. Since the remainder is 0, this tells us that -4 is a zero of the polynomial function. We can also say that $(x + 4)$ is a factor of the polynomial. Also, the other numbers on the bottom row (3, -5, 6) are the coefficients of the quotient, which is $3x^2 - 5x + 6$ . So, the original polynomial can be written as $(x + 4)(3x^2 - 5x + 6)$ .

Now, let's see if 3 is a zero. We'll follow the same process:

Set up:

3 | 3   7  -14  24
    |_________________

Bring down:

3 | 3   7  -14  24
    |_________________
     3

Multiply and add: 3 * 3 = 9. 7 + 9 = 16

3 | 3   7  -14  24
    |    9
    |_________________
     3   16

Repeat: 16 * 3 = 48. -14 + 48 = 34

3 | 3   7  -14  24
    |    9   48
    |_________________
     3   16  34

Repeat again: 34 * 3 = 102. 24 + 102 = 126

3 | 3   7  -14  24
    |    9   48  102
    |_________________
     3   16  34  126

The remainder here is 126. Since the remainder is not 0, this tells us that 3 is not a zero of the polynomial function.

Conclusion: Zeros and Factors

So, to wrap things up, here’s what we found using synthetic division:

-4 is a zero of the function $f(x) = 3x^3 + 7x^2 - 14x + 24$ . Also, $(x + 4)$ is a factor.
3 is not a zero of the function.

Synthetic division is a pretty neat tool, right? It's a quick and efficient way to determine if a given number is a zero of a polynomial. It helps us find the x-intercepts, factor polynomials, and solve polynomial equations. The remainder theorem tells us that if $f(c) = 0$ , then c is a zero of the polynomial $f(x)$ , and $(x - c)$ is a factor. Conversely, if $(x - c)$ is a factor of $f(x)$ , then c is a zero. Knowing these properties helps us understand the structure of the polynomial and its graph. Being able to quickly identify zeros lets us simplify complex functions and explore their characteristics more thoroughly. This is a foundational concept that supports advanced topics in calculus, such as finding extrema and analyzing function behavior.

Keep practicing, and you'll get the hang of it in no time. If you have any more questions, feel free to ask. Happy math-ing, everyone! And remember, synthetic division is your friend! It makes finding the zeros of polynomial functions a breeze. Keep practicing, and you'll become a pro at this. Keep up the good work! And that is how you use synthetic division to find out if the given numbers are zeros of the polynomial function. Isn't math fun? Good luck!