Finding Roots Of Polynomial Expression X(x – 3)(x + 5)(x² + 4) – 6

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Hey guys! Today, we're diving into a fun math problem where we need to figure out which value of x makes the expression x(x – 3)(x + 5)(x² + 4) – 6 equal to zero. It might look a bit intimidating at first, but don't worry, we'll break it down step by step. Let's get started!

Understanding the Problem

So, the big question is: Which of the following values for x can zero the expression x(x – 3)(x + 5)(x² + 4) – 6? We've got these options to choose from:

  • A) -5
  • B) 0
  • C) 3
  • D) 6

To tackle this, we need to understand what it means to “zero” an expression. In mathematical terms, we're looking for the roots or solutions of the equation x(x – 3)(x + 5)(x² + 4) – 6 = 0. These roots are the values of x that make the equation true. Essentially, we are trying to find the values of x that will make the entire expression equal to zero. This involves understanding the structure of the expression and how each part contributes to the final result. The expression consists of several factors multiplied together and then subtracted by 6. To find the roots, we need to determine which values of x will cause the product of the factors to equal 6, thus canceling out the subtraction. This may involve testing each option provided or using algebraic methods to isolate x. Each factor in the expression plays a crucial role. The linear factors (x, x – 3, and x + 5) will equal zero when x is 0, 3, and -5, respectively. The quadratic factor (x² + 4) is always positive because x² is non-negative, and adding 4 keeps it strictly above zero. This means the quadratic factor does not contribute to real roots of the expression. However, it does affect the overall value of the expression, particularly when multiplied by the other factors. The constant term -6 shifts the entire expression downwards. This means that if the product of the factors equals 6, the entire expression will be zero. This shift is crucial for finding the specific values of x that satisfy the equation. By carefully analyzing the expression and considering the effects of each factor and the constant term, we can strategically test the given options to find the root or roots that make the expression equal to zero. This understanding is essential for solving the problem efficiently and accurately.

Breaking Down the Expression

Let's take a closer look at our expression: x(x – 3)(x + 5)(x² + 4) – 6. It's made up of a few key parts:

  1. x: This is a simple variable term. Its value will directly impact the overall expression.
  2. (x – 3): This is a linear factor. It becomes zero when x is 3.
  3. (x + 5): Another linear factor, which is zero when x is -5.
  4. (x² + 4): This is a quadratic factor. Notice that x² will always be positive or zero, and we're adding 4 to it. This means this factor will always be greater than zero (it won't give us any real roots).
  5. – 6: This constant term shifts the whole expression down. Understanding these components is essential. The factors x, (x – 3), and (x + 5) can individually become zero, which would make the entire product zero if it weren't for the – 6. The quadratic factor (x² + 4) is a bit different. Since x² is always non-negative for real numbers (it's either zero or positive), adding 4 makes this factor always positive. This is important because it means (x² + 4) won’t be a source of roots (values that make the expression zero). It will, however, affect the magnitude of the overall product. The – 6 at the end is the key here. To make the whole expression zero, we need the product of all the factors to equal 6. This is because 6 – 6 = 0. So, our goal is to find an x that makes x(x – 3)(x + 5)(x² + 4) equal to 6. This significantly narrows down our approach. We don't need to worry about values that make the product zero; instead, we need values that make the product equal to 6. By focusing on this target value, we can more efficiently test the options given and find the correct answer. This approach simplifies the problem and makes it much more manageable. Breaking down the expression in this way allows us to see the role each part plays and strategize our solution method.

The Strategy: Plugging in the Values

Since we have specific options (A, B, C, and D), the easiest way to solve this is to plug each value of x into the expression and see if we get zero. This is a straightforward method and works well when you have a limited set of potential solutions. This method, often called substitution, is particularly useful when dealing with complex expressions where direct algebraic manipulation might be difficult. By substituting each potential value of x into the expression, we can directly evaluate the result and see if it equals zero. This eliminates the need for intricate factoring or solving higher-order equations. It's a practical and efficient approach, especially in multiple-choice scenarios where the answer is one of the given options. When plugging in the values, it's essential to follow the order of operations (PEMDAS/BODMAS) to ensure accurate calculations. This means performing operations in the correct sequence: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Skipping or misinterpreting this order can lead to incorrect results. For instance, in our expression, we need to evaluate the factors inside the parentheses first, then perform the multiplication, and finally, the subtraction. It's also beneficial to keep track of the calculations neatly, perhaps using a separate piece of paper, to avoid errors. Write down each step and the result to make it easier to review and correct any mistakes. By following these tips, we can confidently use the substitution method to solve the problem, systematically testing each option until we find the one that zeros the expression.

Testing the Options

Let's go through each option one by one:

A) x = -5

Plug in x = -5 into the expression:

(-5)((-5) – 3)((-5) + 5)((-5)² + 4) – 6

(-5)(-8)(0)(29) – 6

The product becomes zero because of the (0) factor:

0 – 6 = -6

So, x = -5 doesn't work.

B) x = 0

Plug in x = 0:

(0)((0) – 3)((0) + 5)((0)² + 4) – 6

(0)(-3)(5)(4) – 6

Again, the product is zero:

0 – 6 = -6

So, x = 0 doesn't work either.

C) x = 3

Plug in x = 3:

(3)((3) – 3)((3) + 5)((3)² + 4) – 6

(3)(0)(8)(13) – 6

Once more, the product is zero:

0 – 6 = -6

So, x = 3 is not the solution.

D) x = 6

Let's try x = 6:

(6)((6) – 3)((6) + 5)((6)² + 4) – 6

(6)(3)(11)(40) – 6

7920 – 6 = 7914

This doesn't equal zero, so x = 6 is not the answer.

Whoops! It seems like there was a small oversight in the original expression. The expression is supposed to be x(x – 3)(x + 5)(x² + 4) – 6, but let's correct it to x(x – 3)(x + 5)(x² + 4) - 2 - 4 which simplifies to x(x – 3)(x + 5)(x² + 4) – 6. We already calculated using the correct expression. It appears none of the provided options make the expression equal to zero. This often happens in math problems! Sometimes, the answer isn't in the given choices, or there might be a slight mistake in the problem itself. It's a good reminder always to double-check your work and the problem statement. In real-world scenarios, it is important to recognize when a problem might have no solution within the constraints given, which can lead to re-evaluating the approach or the initial assumptions. This situation also highlights the importance of critical thinking in problem-solving. When the expected outcome isn't achieved, instead of just accepting the result, it's crucial to pause and analyze why. This involves reviewing the steps taken, the calculations made, and even the initial problem statement. In this case, our systematic approach of substituting values helped us confirm that none of the options work. But it also prompted us to reconsider the possibility of an error in the problem itself. This analytical mindset is key to effective problem-solving and is a valuable skill in any field. Remember, guys, math isn't just about finding the right answer; it's also about the process of thinking, questioning, and analyzing!

The Corrected Analysis and Conclusion

Okay, so it looks like we've got a bit of a twist! After carefully testing each option, we found that none of them actually zero the expression x(x – 3)(x + 5)(x² + 4) – 6. This is a great learning moment because it shows us that sometimes, the answer isn't always neatly within the given choices. Instead of panicking, let's think about what this means. It could suggest a couple of things:

  1. There might be a slight error in the problem or the options provided. This happens more often than you think! Math problems, especially in practice scenarios, can sometimes have typos or mistakes.
  2. The actual roots of the equation might be numbers that aren't in our options. The roots could be decimals, fractions, or even imaginary numbers (involving the square root of -1), which aren't covered by our simple integer options.

So, what should we take away from this? First, always double-check your work. We did that by carefully plugging in each option and making sure we followed the order of operations. Second, don't be afraid to question the problem itself. If you've done everything correctly and the answer still doesn't fit, it's okay to consider the possibility of an error. Guys, this is a crucial skill in math and life in general! It's about critical thinking and not just blindly following steps. Sometimes, the most valuable lesson is learning to recognize when something isn't quite right and to think creatively about why. In conclusion, for the given options and the expression x(x – 3)(x + 5)(x² + 4) – 6, none of the values A) -5, B) 0, C) 3, or D) 6 will zero the expression. This highlights the importance of thorough checking and the possibility that not all problems have simple, straightforward solutions within the given constraints.