Finding Factors And Zeros Of Polynomials: A Step-by-Step Guide

by TextBrain Team 63 views

Let's dive into the world of polynomials, guys! Today, we're tackling a problem where we're given a zero of a polynomial function and asked to find the corresponding factor, its multiplicity, and the potential rational zeros according to the Rational Root Theorem. Sounds like a mouthful, but we'll break it down step by step. So, buckle up and let's get started!

Identifying the Factor and Its Multiplicity

Okay, first things first, we know that x = 3 is a zero of the polynomial function h(x) = -2x⁴ - 63x² + 19x³ + 81x - 27. But what does that even mean? Simply put, if a value is a zero of a function, it means that plugging that value into the function makes the function equal to zero. And more importantly for us, it tells us something about the factors of the polynomial.

If x = 3 is a zero, then (x - 3) must be a factor of h(x). This is a fundamental concept in polynomial algebra. Think of it like this: if a number makes a polynomial equal to zero, then subtracting that number from x will give you a factor. Easy peasy, right?

Now, the tricky part: What's the multiplicity of this factor? The multiplicity tells us how many times this factor appears in the fully factored form of the polynomial. It's like asking, "Is (x - 3) just a factor once, or does it show up multiple times?" To figure this out, we're going to use a technique called synthetic division.

Synthetic Division: A Quick Refresher

Synthetic division is a neat and efficient way to divide a polynomial by a linear factor like (x - 3). It's basically a shortcut version of long division, but way less messy. Here's how it works:

  1. Write down the coefficients of the polynomial. Make sure you include zeros as placeholders for any missing terms. In our case, h(x) = -2x⁴ + 19x³ - 63x² + 81x - 27, so our coefficients are -2, 19, -63, 81, and -27.
  2. Write the zero (which is 3) to the left.
  3. Draw a line below the coefficients.
  4. Bring down the first coefficient (-2) below the line.
  5. Multiply the zero (3) by the number you just brought down (-2), which gives you -6. Write this below the next coefficient (19).
  6. Add the numbers in that column (19 + (-6) = 13). Write the result (13) below the line.
  7. Repeat steps 5 and 6 for the remaining coefficients.

If you've done it correctly, you'll end up with a row of numbers below the line. The last number is the remainder, and the other numbers are the coefficients of the quotient polynomial.

Performing Synthetic Division

Let's do the synthetic division for h(x) with x = 3:

3 | -2 19 -63 81 -27
    -6 39 -72 27
   -------------------------
   -2 13 -24 9 0

Notice that the remainder is 0, which confirms that (x - 3) is indeed a factor of h(x). The other numbers, -2, 13, -24, and 9, are the coefficients of the quotient polynomial, which is -2x³ + 13x² - 24x + 9.

Checking for Multiplicity

But we're not done yet! To find the multiplicity, we need to see if (x - 3) is a factor of this new quotient polynomial as well. So, let's do synthetic division again, using the quotient's coefficients:

3 | -2 13 -24 9
    -6 21 -9
   ----------------
   -2 7 -3 0

Again, the remainder is 0! This means (x - 3) is also a factor of -2x³ + 13x² - 24x + 9. Our new quotient polynomial is -2x² + 7x - 3.

Let's try one more time:

3 | -2 7 -3
    -6 3
   ------------
   -2 1 0

The remainder is still 0! This means (x - 3) is also a factor of -2x² + 7x - 3. The newest quotient polynomial is -2x + 1.

Now, if we try synthetic division one more time:

3 | -2 1
    -6
   ------
   -2 -5

The remainder is no longer 0. This tells us that (x - 3) is not a factor of -2x + 1. So, we've hit the limit!

The Verdict on Multiplicity

We successfully divided by (x - 3) three times, each time getting a remainder of 0. This means that (x - 3) is a factor of h(x) with a multiplicity of 3. It appears three times in the fully factored form of the polynomial.

Applying the Rational Root Theorem

Alright, now let's switch gears and talk about the Rational Root Theorem. This theorem is a super handy tool for figuring out the potential rational zeros of a polynomial. It doesn't tell us exactly what the zeros are, but it gives us a list of possible candidates to test.

What is the Rational Root Theorem?

The Rational Root Theorem states that if a polynomial has integer coefficients, then any rational zero of the polynomial must be of the form p/q, where p is a factor of the constant term (the term without any x in it) and q is a factor of the leading coefficient (the coefficient of the term with the highest power of x).

In simpler terms:

  • List all the factors of the constant term (both positive and negative).
  • List all the factors of the leading coefficient (both positive and negative).
  • Create all possible fractions by dividing each factor of the constant term by each factor of the leading coefficient. These fractions are your potential rational zeros.

Applying the Theorem to Our Polynomial

Let's apply this to our polynomial, h(x) = -2x⁴ + 19x³ - 63x² + 81x - 27.

  1. The constant term is -27. Its factors are ±1, ±3, ±9, and ±27.
  2. The leading coefficient is -2. Its factors are ±1 and ±2.
  3. Now, let's create the possible fractions (p/q):
    • ±1/1 = ±1
    • ±3/1 = ±3
    • ±9/1 = ±9
    • ±27/1 = ±27
    • ±1/2
    • ±3/2
    • ±9/2
    • ±27/2

So, according to the Rational Root Theorem, there are 16 possible rational zeros for this function. Remember, these are just potential zeros. Some of them might not actually be zeros, and there might be irrational or complex zeros as well. But the theorem gives us a great starting point for finding the rational zeros.

Wrapping It Up

Today, we've explored how to find the factor corresponding to a given zero, determine its multiplicity using synthetic division, and apply the Rational Root Theorem to identify potential rational zeros. These are powerful tools in your polynomial-solving arsenal, guys!

Remember, practice makes perfect. The more you work with these concepts, the more comfortable you'll become. So, keep practicing, and you'll be a polynomial pro in no time! You got this! If you found this helpful, feel free to share this guide with your friends and classmates. Happy solving!