Finding All Zeros Of A Cubic Function: A Step-by-Step Guide

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Hey everyone! Today, we're diving into the world of polynomial functions and figuring out how to find all the zeros of a cubic function. Let's take the example of f(x) = 2x³ + 3x² + 8x + 12. We're given a little head start: 2i is one of the zeros. Don't worry if complex numbers make you sweat; we'll break it down piece by piece. It is important to understand this concept since polynomials are fundamental in mathematics, popping up everywhere from calculus to physics. Being able to find zeros (or roots) is a key skill. You can use it to solve various problems, like determining the points where a function crosses the x-axis or modeling real-world scenarios. Ready to become a zero-finding master? Let’s get started.

Understanding the Problem and the Tools

First things first, what exactly are we trying to do? The zeros of a function are the x-values where the function equals zero, i.e., where f(x) = 0. Think of it as finding the x-intercepts on a graph. In our case, we're dealing with a cubic function. A cubic function has the general form ax³ + bx² + cx + d = 0, where a, b, c, and d are constants. Because it's a cubic function, we know, based on the Fundamental Theorem of Algebra, that there should be three zeros (counting multiplicity). Now, our problem gives us a complex zero, 2i. Remember that complex numbers come in the form a + bi, where a is the real part, and b is the imaginary part, and i is the imaginary unit (i² = -1). When a complex number is a zero of a polynomial with real coefficients (which our function has), its complex conjugate is also a zero. This is a super helpful fact because it gives us two zeros right away!

We’ll be using a couple of key techniques here. First, we'll use the Complex Conjugate Root Theorem. This theorem tells us that if a + bi is a zero of a polynomial, then a - bi is also a zero. Second, we'll apply polynomial division to simplify our cubic function into a quadratic one, which is much easier to solve. Finally, we will use the quadratic formula to find the remaining zeros. The quadratic formula is your best friend when it comes to solving quadratic equations, giving you the ability to solve any quadratic equation.

Step 1: Finding the Conjugate and Forming a Quadratic Factor

Alright, guys, let's roll up our sleeves! Since 2i is a zero, and knowing the Complex Conjugate Root Theorem, we know that the complex conjugate, -2i, is also a zero. Great, we've got two zeros already! This means that (x - 2i) and (x + 2i) are factors of our cubic function. Now, let's multiply these two factors together to get a quadratic factor. Doing so will make our next step more manageable.

So, (x - 2i)(x + 2i) = x² - (2i)² = x² - 4i². Remember, i² = -1, so we can simplify this to x² - 4(-1) = x² + 4. We now have a quadratic factor: x² + 4. This factor will be used in the next step: polynomial division. What's awesome is that this factor will always give you a real quadratic. In our scenario, the other factor is x² + 4, which is a sum of squares. The sum of squares can only be factored using complex numbers, which is what we started with!

Step 2: Polynomial Division

Time to use polynomial division! We're going to divide our original cubic function, 2x³ + 3x² + 8x + 12, by the quadratic factor we just found, x² + 4. It might seem a little daunting if you've not done it in a while, but trust me, it’s totally doable. Polynomial division allows us to break down the cubic function into simpler factors. Set it up like a regular long division problem. Let's work through it step by step:

  1. Divide the first term of the dividend (2x³) by the first term of the divisor (x²). This gives us 2x. Write this above the division symbol.
  2. Multiply the divisor (x² + 4) by 2x. This gives us 2x³ + 8x. Write this below the dividend.
  3. Subtract the result from the dividend. This eliminates the 2x³ term and leaves us with 3x² + 12. Remember to distribute the negative sign correctly!
  4. Bring down the remaining term (12) and continue this process. Divide 3x² by x², which gives us 3. Write this above the division symbol.
  5. Multiply the divisor (x² + 4) by 3. This gives us 3x² + 12. Write this below the remaining terms.
  6. Subtract. This gives us a remainder of 0. Yay! This means our division worked perfectly.

After the division, we get the quotient 2x + 3. This means that 2x³ + 3x² + 8x + 12 = (x² + 4)(2x + 3). That is, we've successfully factored our cubic function. Now we can use the factors to find the zeros.

Step 3: Finding the Remaining Zero

We are almost there! Now that we've factored the cubic function into (x² + 4)(2x + 3), it's easy to find the remaining zero. We already know the zeros from (x² + 4) are 2i and -2i because they are complex conjugates. So, we just need to solve for the zero from the factor (2x + 3).

Set (2x + 3 = 0) and solve for x. Subtract 3 from both sides: 2x = -3. Divide both sides by 2: x = -3/2. Therefore, the remaining zero is -3/2. This zero is a real number, in contrast to the complex zeros we found earlier. Real and complex zeros work together, and it all makes sense in the grand scheme of the polynomial function.

Step 4: Listing All the Zeros

We did it, guys! We've successfully found all the zeros of the cubic function f(x) = 2x³ + 3x² + 8x + 12. The zeros are:

  • 2i (given)
  • -2i (the complex conjugate)
  • -3/2 (found by solving 2x + 3 = 0)

And there you have it! We’ve used our knowledge of complex conjugates and polynomial division to solve this cubic function. Remember, practice makes perfect. So try more examples. The more you practice, the better you'll become at solving these types of problems. You're well on your way to mastering polynomials! Keep up the great work, and don’t be afraid to ask questions. Math can be fun when you have the right approach!