Factoring Polynomials: Find Zeros & Multiplicity

Let's dive into the world of polynomial factorization! In this article, we're going to break down how to completely factor a polynomial, find its zeros, and understand the multiplicity of each zero. We'll focus on the polynomial O(x) = x^2 - 4x + 5, walking through each step in detail so you can tackle similar problems with confidence. So, buckle up, math enthusiasts, and let’s get started!

Factoring the Polynomial O(x) = x^2 - 4x + 5

When we're faced with a polynomial like O(x) = x^2 - 4x + 5, the first thing we want to do is try to factor it. Factoring means expressing the polynomial as a product of simpler polynomials. This can often make it easier to find the zeros (the values of x for which the polynomial equals zero). For a quadratic polynomial like this, we typically look for two binomials that multiply together to give us the original polynomial. To start factoring the polynomial O(x) = x^2 - 4x + 5, we need to look for two numbers that multiply to 5 (the constant term) and add up to -4 (the coefficient of the x term). However, in this case, we quickly realize that there are no two real numbers that satisfy these conditions. The factors of 5 are 1 and 5, or -1 and -5. Neither of these pairs adds up to -4. This suggests that the polynomial might not factor nicely using real numbers.

Since simple factoring isn't working, the next logical step is to check the discriminant. The discriminant, often denoted as Δ (Delta), is a part of the quadratic formula that tells us about the nature of the roots (zeros) of the quadratic equation. The formula for the discriminant is Δ = b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation in the standard form ax^2 + bx + c = 0. In our case, O(x) = x^2 - 4x + 5, so a = 1, b = -4, and c = 5. Plugging these values into the discriminant formula, we get: Δ = (-4)^2 - 4 * 1 * 5 = 16 - 20 = -4. A negative discriminant means that the quadratic equation has two complex roots (zeros). This confirms our suspicion that the polynomial doesn't factor into real binomials. Because the discriminant is negative, we know that the zeros will be complex numbers, involving the imaginary unit 'i', where i is the square root of -1. So, we'll need to use the quadratic formula to find these complex zeros, which will then help us express the polynomial in its completely factored form over the complex numbers.

Finding the Zeros Using the Quadratic Formula

Since we determined that the polynomial O(x) = x^2 - 4x + 5 has complex roots, we need to use the quadratic formula to find these zeros. The quadratic formula is a powerful tool that provides the solutions (zeros) for any quadratic equation in the form ax^2 + bx + c = 0. The formula is given by: x = (-b ± √(b^2 - 4ac)) / (2a). We already know the values of a, b, and c for our polynomial: a = 1, b = -4, and c = 5. We also calculated the discriminant, Δ = b^2 - 4ac = -4, in the previous section. Now, we can plug these values into the quadratic formula to find the zeros.

Substituting the values, we get: x = (-(-4) ± √((-4)^2 - 4 * 1 * 5)) / (2 * 1) which simplifies to x = (4 ± √(-4)) / 2. The square root of -4 can be expressed using the imaginary unit 'i' as √(-4) = √(4 * -1) = √(4) * √(-1) = 2i. So, our equation becomes x = (4 ± 2i) / 2. Now, we can simplify this expression by dividing both the real and imaginary parts by 2: x = 2 ± i. This gives us two complex zeros: x1 = 2 + i and x2 = 2 - i. These are the values of x that make the polynomial O(x) equal to zero. Notice that these zeros are complex conjugates, which is a common occurrence when dealing with quadratic equations that have a negative discriminant. Finding these zeros is a crucial step in completely factoring the polynomial, as we'll see in the next section. Now that we have the zeros, we can use them to write the polynomial in its factored form.

Stating the Multiplicity of Each Zero

Now that we've found the zeros of the polynomial O(x) = x^2 - 4x + 5, the next step is to state the multiplicity of each zero. The multiplicity of a zero refers to the number of times that zero appears as a root of the polynomial. In simpler terms, it's how many times the corresponding factor appears in the factored form of the polynomial. For example, if a polynomial has a factor of (x - 2)^3, then the zero x = 2 has a multiplicity of 3, meaning it appears as a root three times. In our case, we found two complex zeros: x1 = 2 + i and x2 = 2 - i. Since these zeros were obtained from the quadratic formula and each appears only once as a solution, their multiplicity is 1. This is typical for quadratic equations, which have a degree of 2 and thus have two roots (zeros), counting multiplicity. If a quadratic had a repeated root (meaning the discriminant was zero), that root would have a multiplicity of 2.

In the context of polynomial graphs, the multiplicity of a zero tells us about the behavior of the graph near that zero. If a zero has a multiplicity of 1, the graph crosses the x-axis at that point. If a zero has a multiplicity of 2, the graph touches the x-axis at that point but doesn't cross it (it bounces off). For higher multiplicities, the graph may have a more complex shape near the zero. However, since our zeros are complex, they don't appear on the real number line (x-axis), so this graphical interpretation doesn't directly apply. In summary, for the polynomial O(x) = x^2 - 4x + 5, the zeros are x1 = 2 + i and x2 = 2 - i, and each has a multiplicity of 1. This completes our analysis of the zeros and their multiplicities for this polynomial. Understanding multiplicity is crucial for sketching polynomial graphs and for solving more advanced polynomial problems.

Ordering the Answers

The final part of the problem asks us to order the answers from smallest to largest real, followed by complex answers ordered by smallest to largest real part, then smallest to largest imaginary part. This is an important step to ensure that we present our solutions in a clear and organized manner. Ordering real numbers is straightforward, as we can simply compare their values on the number line. However, ordering complex numbers requires a bit more consideration, as they have both real and imaginary parts. In our case, we have two complex zeros: x1 = 2 + i and x2 = 2 - i. To order these complex numbers, we first compare their real parts. Both zeros have a real part of 2, so we move on to comparing their imaginary parts.

The imaginary part of x1 = 2 + i is 1, and the imaginary part of x2 = 2 - i is -1. Since -1 is smaller than 1, we order the zeros as x2 = 2 - i, followed by x1 = 2 + i. Therefore, the zeros of the polynomial O(x) = x^2 - 4x + 5, ordered according to the problem's instructions, are 2 - i and 2 + i. It's worth noting that complex zeros of polynomials with real coefficients always come in conjugate pairs (a + bi and a - bi), which simplifies the ordering process since their real parts are the same. This ordering convention is useful in various mathematical contexts, especially when dealing with polynomial roots and solutions to equations. By following this systematic approach, we can confidently present our solutions in the correct order.

Conclusion

Alright, guys, we've successfully factored the polynomial O(x) = x^2 - 4x + 5, found its zeros, stated their multiplicities, and ordered them according to the instructions. We started by attempting to factor the polynomial using real numbers, but quickly realized that it had complex roots. We then used the discriminant to confirm this and applied the quadratic formula to find the complex zeros: 2 + i and 2 - i. Each of these zeros has a multiplicity of 1, meaning they each appear once as a root of the polynomial. Finally, we ordered the zeros, taking into account their real and imaginary parts. This comprehensive process demonstrates how to tackle polynomial factorization and root-finding, even when dealing with complex numbers. Remember, practice makes perfect, so keep working on these types of problems to build your skills and confidence! Factoring polynomials and finding their zeros is a fundamental skill in algebra and calculus, so mastering these concepts will set you up for success in more advanced math courses. Keep up the great work!