Finding All Roots Of F(x) = X³ + 10x² - 25x - 250 Using The Remainder Theorem
Hey guys! Today, we're diving into the fascinating world of polynomial functions and their roots. Specifically, we're going to tackle the function f(x) = x³ + 10x² - 25x - 250. We already know one root, which is x = -10, and our mission is to find the remaining roots. We'll be using the Remainder Theorem, a powerful tool in polynomial algebra, to help us on this quest. So, buckle up and let's get started!
Understanding the Remainder Theorem
Before we jump into the calculations, let's quickly recap what the Remainder Theorem is all about. Simply put, the Remainder Theorem states that if you divide a polynomial f(x) by (x - c), the remainder is equal to f(c). In other words, if you plug c into the polynomial, the result you get is the same as the remainder when you divide by (x - c). This theorem is super handy for finding roots because if f(c) = 0, then c is a root of the polynomial. This is a cornerstone concept for us, and we'll use it to confirm our known root and to simplify our polynomial for further root-finding adventures.
When we talk about finding roots, we're essentially looking for the values of x that make f(x) = 0. These roots are also known as the zeros of the polynomial. For a cubic polynomial like the one we're dealing with, there can be up to three roots (though some might be repeated or complex). Knowing one root is like having a head start in a race – it gives us a significant advantage in finding the others. In our case, knowing that x = -10 is a root means that (x + 10) is a factor of our polynomial. This is a key piece of information that we'll use to break down our cubic polynomial into smaller, more manageable pieces. Factoring a polynomial is like breaking down a complex problem into smaller, simpler sub-problems, which is a common and effective strategy in mathematics.
Applying the Remainder Theorem with Synthetic Division
Now that we've got the Remainder Theorem fresh in our minds, let's put it into action! We know that x = -10 is a root of f(x) = x³ + 10x² - 25x - 250. This means that if we divide f(x) by (x + 10), the remainder should be zero. To perform this division efficiently, we'll use a technique called synthetic division. Synthetic division is a streamlined way to divide a polynomial by a linear factor, and it's especially useful when we're trying to find roots.
Here's how synthetic division works in our case:
- Write down the coefficients of the polynomial: 1, 10, -25, -250.
- Write down the root we know, which is -10, on the left.
- Bring down the first coefficient (1) below the line.
- Multiply the root (-10) by the number you just brought down (1) and write the result (-10) under the second coefficient (10).
- Add the second coefficient (10) and the result you just wrote (-10), which gives you 0. Write this below the line.
- Multiply the root (-10) by the number you just got (0) and write the result (0) under the third coefficient (-25).
- Add the third coefficient (-25) and the result you just wrote (0), which gives you -25. Write this below the line.
- Multiply the root (-10) by the number you just got (-25) and write the result (250) under the last coefficient (-250).
- Add the last coefficient (-250) and the result you just wrote (250), which gives you 0. Write this below the line.
The last number below the line (0) is the remainder. As expected, it's zero, which confirms that x = -10 is indeed a root. The other numbers below the line (1, 0, -25) are the coefficients of the quotient polynomial. In this case, the quotient is x² + 0x - 25, which simplifies to x² - 25. Synthetic division not only confirms our root but also helps us reduce a cubic equation into a quadratic one, making it far easier to solve.
Solving the Quadratic Equation
After performing synthetic division, we've successfully reduced our cubic polynomial f(x) into (x + 10)(x² - 25). This is a fantastic achievement because we've essentially broken down a complex problem into simpler parts. We already know that x = -10 is a root from the (x + 10) factor. Now, we need to find the roots of the quadratic factor x² - 25. This quadratic equation is a classic example of a difference of squares, a pattern that's worth recognizing and remembering.
The difference of squares pattern states that a² - b² = (a + b)(a - b). In our case, x² - 25 fits this pattern perfectly, with a = x and b = 5. Therefore, we can factor x² - 25 as (x + 5)(x - 5). Now, our polynomial f(x) is fully factored as (x + 10)(x + 5)(x - 5).
To find the roots, we simply set each factor equal to zero and solve for x:
- x + 10 = 0 => x = -10
- x + 5 = 0 => x = -5
- x - 5 = 0 => x = 5
Therefore, the roots of the function f(x) = x³ + 10x² - 25x - 250 are x = -10, x = -5, and x = 5. We've successfully found all the roots using the Remainder Theorem and the difference of squares pattern. Isn't it satisfying when a plan comes together?
Conclusion: All Roots Revealed
Alright, guys, we've reached the end of our root-finding journey! We started with a cubic polynomial, f(x) = x³ + 10x² - 25x - 250, and the knowledge that x = -10 was one of its roots. Using the Remainder Theorem and synthetic division, we skillfully broke down the polynomial and discovered the remaining roots. We found that the other roots are x = -5 and x = 5. So, the complete set of roots for the function is x = -10, x = -5, and x = 5.
This exercise highlights the power of the Remainder Theorem and factoring techniques in solving polynomial equations. By understanding and applying these concepts, you can confidently tackle similar problems and unravel the mysteries hidden within polynomial functions. Remember, finding roots is a fundamental skill in algebra, and mastering it opens doors to more advanced mathematical concepts. Keep practicing, keep exploring, and keep those mathematical gears turning! You've got this!
