Finding All Roots Of F(x) = (x^2 + 2x - 15)(x^2 + 8x + 17)

Polynomial functions, those mathematical expressions with variables raised to various powers, can seem daunting. But don't worry, guys! We're going to break down how to find all the roots (or solutions) of a polynomial function. In this guide, we'll specifically tackle the function $f(x) = (x^2 + 2x - 15)(x^2 + 8x + 17)$. By the end, you'll be a root-finding pro!

Understanding Roots and Polynomial Functions

Before we dive into the nitty-gritty, let's establish a solid foundation. Polynomial functions are expressions involving variables raised to non-negative integer powers, combined with coefficients and constants. Think of examples like $x^2 + 3x + 2$ or $2x^3 - x + 5$. The roots of a polynomial function are the values of x that make the function equal to zero. In other words, they are the points where the graph of the function intersects the x-axis. Finding these roots is crucial in many areas of mathematics, science, and engineering.

Why are roots so important? Well, they tell us a lot about the function's behavior. They help us understand where the function crosses the x-axis, where it changes direction, and its overall shape. Roots are also fundamental in solving equations, modeling real-world phenomena, and designing systems. In essence, finding the roots unlocks a deeper understanding of the polynomial function itself.

For quadratic equations (polynomials of degree 2), we have the handy quadratic formula, but for higher-degree polynomials, things can get trickier. That's where techniques like factoring and complex numbers come into play. Factoring helps us break down the polynomial into simpler expressions, making it easier to find the roots. Complex numbers, which include an imaginary part, allow us to find roots that aren't on the real number line. This is where things can get really interesting!

So, let's keep these concepts in mind as we move forward. We'll be using factoring, the quadratic formula, and complex numbers to find all the roots of our given polynomial function. Remember, the goal is not just to find the answers but to understand why we're using each step. This way, you'll be able to tackle any polynomial root-finding challenge that comes your way. Think of it like unlocking a secret code – once you know the rules, you can decipher anything!

Factoring the First Quadratic Expression: $x^2 + 2x - 15$

Let's start by tackling the first part of our function, the quadratic expression $x^2 + 2x - 15$. Factoring is a powerful technique that allows us to rewrite a polynomial as a product of simpler expressions. If we can factor this quadratic, we can easily find two of the roots. Remember, the goal of factoring is to find two binomials (expressions with two terms) that multiply together to give us the original quadratic. So, how do we do it?

We need to find two numbers that multiply to -15 (the constant term) and add up to 2 (the coefficient of the x term). Think of it like a puzzle! Let's list the factors of -15: (-1, 15), (1, -15), (-3, 5), and (3, -5). Which of these pairs adds up to 2? Bingo! It's 5 and -3. This means we can rewrite our quadratic expression as follows:

$x^2 + 2x - 15 = (x + 5)(x - 3)$

See how we used the numbers 5 and -3 in our factors? Now, to find the roots from these factors, we use the Zero Product Property. This property states that if the product of two factors is zero, then at least one of the factors must be zero. In other words, if (x + 5)(x - 3) = 0, then either (x + 5) = 0 or (x - 3) = 0 (or both!).

Let's solve each of these equations separately:

• x + 5 = 0 => x = -5
• x - 3 = 0 => x = 3

Great! We've found two roots: -5 and 3. These are the values of x that make the first part of our function equal to zero. But remember, we're not done yet! We still need to deal with the second quadratic expression, $x^2 + 8x + 17$. Factoring this one is a little trickier, as we'll see in the next section. But for now, let's celebrate our success in factoring the first quadratic and finding two of the roots. We're well on our way to solving the entire puzzle!

Tackling the Second Quadratic: $x^2 + 8x + 17$ and the Quadratic Formula

Alright, let's move on to the second quadratic expression in our function: $x^2 + 8x + 17$. This one isn't as easily factorable as the first one. We won't find two nice, whole numbers that multiply to 17 and add up to 8. So, what do we do? This is where the quadratic formula comes to our rescue! The quadratic formula is a powerful tool that allows us to find the roots of any quadratic equation, no matter how complex.

The quadratic formula is given by:

$x = \frac{-b ± \sqrt{b^2 - 4ac}}{2a}$

where a, b, and c are the coefficients of the quadratic equation in the standard form $ax^2 + bx + c = 0$. In our case, for the expression $x^2 + 8x + 17$, we have a = 1, b = 8, and c = 17. Let's plug these values into the quadratic formula:

$x = \frac{-8 ± \sqrt{8^2 - 4(1)(17)}}{2(1)}$

Now, let's simplify this step by step. First, let's calculate the expression under the square root, which is called the discriminant:

$8^2 - 4(1)(17) = 64 - 68 = -4$

Uh oh! We have a negative number under the square root. What does this mean? It means that the roots of this quadratic are not real numbers. They are complex numbers, which involve the imaginary unit i, where i is defined as the square root of -1 ($\sqrt{-1}$). Don't worry, this is perfectly normal! Complex roots arise frequently in mathematics and have important applications in various fields.

Now, let's continue simplifying our expression:

$x = \frac{-8 ± \sqrt{-4}}{2} = \frac{-8 ± 2i}{2}$

We can simplify this further by dividing both terms in the numerator by 2:

$x = -4 ± i$

So, we have two complex roots: -4 + i and -4 - i. These roots are complex conjugates of each other, meaning they have the same real part (-4) but opposite imaginary parts (+i and -i). Complex roots always come in conjugate pairs for polynomials with real coefficients, which is a neat fact to remember!

We've successfully navigated the quadratic formula and found the complex roots of our second quadratic expression. This shows the power of the quadratic formula in handling cases where factoring isn't straightforward. Now, we're just one step away from finding the complete list of roots for our original polynomial function.

The Complete List of Roots: Putting It All Together

Okay, guys, we've done the hard work! We've factored the first quadratic expression, used the quadratic formula for the second, and now it's time to put all the pieces together. Remember, our original function was:

$f(x) = (x^2 + 2x - 15)(x^2 + 8x + 17)$

We found the roots of the first quadratic, $x^2 + 2x - 15$, to be -5 and 3. And we found the roots of the second quadratic, $x^2 + 8x + 17$, to be -4 + i and -4 - i. So, the complete list of roots for the polynomial function f(x) is simply the combination of all these roots:

-5, 3, -4 + i, -4 - i

That's it! We've found all the values of x that make the function f(x) equal to zero. We've successfully navigated factoring, the quadratic formula, and even complex numbers. Give yourselves a pat on the back!

What does this list of roots tell us about the polynomial function? Well, the real roots (-5 and 3) represent the points where the graph of the function crosses the x-axis. The complex roots (-4 + i and -4 - i) don't show up on the real number line, but they are still crucial for understanding the function's overall behavior. They indicate the function has a certain symmetry and curvature that wouldn't be apparent if we only considered the real roots.

So, we've not only found the roots but also gained a deeper appreciation for what they represent. This is the power of understanding polynomial functions and their roots. By using techniques like factoring and the quadratic formula, we can unlock the secrets hidden within these expressions. Keep practicing, and you'll become a master of root-finding in no time!

Conclusion: Mastering Polynomial Roots

In this comprehensive guide, we've explored how to find the complete list of roots for the polynomial function $f(x) = (x^2 + 2x - 15)(x^2 + 8x + 17)$. We started by understanding the fundamental concepts of polynomial functions and roots, emphasizing their importance in mathematics and various applications. We then walked through the process step by step, from factoring the first quadratic expression to applying the quadratic formula to find complex roots for the second quadratic.

We've seen how factoring can simplify the process of finding roots when applicable, and we've also learned how the quadratic formula acts as a universal tool for solving any quadratic equation, even those with complex roots. The appearance of complex roots highlights the richness and complexity of polynomial functions, and understanding them is crucial for a complete understanding of the function's behavior.

Remember, the key to mastering polynomial roots is practice. The more you work with different polynomial functions, the more comfortable you'll become with the techniques involved. Don't be afraid to tackle challenging problems, and always strive to understand the why behind each step. With perseverance and a solid understanding of the concepts, you'll be able to confidently find the roots of any polynomial function that comes your way. So, keep exploring, keep learning, and keep unlocking the fascinating world of mathematics!