Synthetic Division Is K=1 A Zero Of P(x)=3x⁴-9x³-8x²+28x-14

Hey guys! Today, we're diving into the world of polynomials and synthetic division. We'll tackle the question of whether a given value, k, is a zero of a polynomial and, if it's not, how to find p(k). Specifically, we're looking at the polynomial p(x) = 3x⁴ - 9x³ - 8x² + 28x - 14 and checking if k = 1 is a zero. So, let's roll up our sleeves and get started!

Understanding Polynomial Zeros

First, let's clarify what we mean by a "zero" of a polynomial. A zero (also sometimes called a root) of a polynomial p(x) is a value k such that p(k) = 0. In simpler terms, if you plug k into the polynomial, the result is zero. Finding zeros is a crucial part of working with polynomials, as they tell us a lot about the polynomial's behavior, including where it crosses the x-axis on a graph. There are several ways to find the zeros of a polynomial, but one of the most efficient methods, especially when you want to test a specific value, is synthetic division.

Synthetic division is a streamlined way of dividing a polynomial by a linear factor of the form (x - k). It's a shorthand version of polynomial long division, and it's incredibly useful for determining both if k is a zero and, if it's not, what the remainder is when you divide by (x - k). The remainder we get from synthetic division is precisely the value of p(k), thanks to the Remainder Theorem. So, if the remainder is zero, k is a zero of the polynomial. If the remainder is not zero, it's the value of p(k). In our case, we are testing if k=1 is a zero of p(x)=3x⁴-9x³-8x²+28x-14, which makes synthetic division the perfect tool for the job.

Setting up Synthetic Division for p(x) with k=1

Now, let’s walk through the steps of using synthetic division for our polynomial p(x) = 3x⁴ - 9x³ - 8x² + 28x - 14 with k = 1. Synthetic division might seem a bit like a magic trick at first, but it’s a straightforward process once you understand the mechanics. First, we need to set up our synthetic division table. Write down the coefficients of the polynomial in a row. Make sure you include a zero for any missing terms (e.g., if there was no x term, we’d include a 0). In our case, the coefficients are 3, -9, -8, 28, and -14. These correspond to the coefficients of x⁴, x³, x², x, and the constant term, respectively. Next, we write the value of k (which is 1 in our case) to the left of these coefficients. We'll draw a vertical line to separate k from the coefficients and a horizontal line below the coefficients where we'll write the results of our calculations. So far, our setup looks something like this:

1 | 3 -9 -8 28 -14
  |________________________
  |

This sets the stage for the magic to happen. Remember, it's crucial to get the signs of the coefficients correct, as a small mistake can throw off the whole process. Now that our table is set up, we can begin the actual synthetic division process. The first step is simple: just bring down the first coefficient (which is 3 in our case) below the horizontal line. This number is the first coefficient of our quotient (we’ll talk more about the quotient later), but for now, just think of it as the starting point for our calculations.

Performing the Synthetic Division

Alright, let's get into the heart of the synthetic division process. We've already set up our table and brought down the first coefficient. Now comes the repetitive part, which is where the magic really happens. We're going to multiply, add, multiply, add, and so on until we reach the end of the line. The key is to take it one step at a time and keep track of your numbers. The general rule is that the first number under the line is multiplied by k. So, we take the 3 that we brought down earlier and multiply it by k = 1. This gives us 3 * 1 = 3. We then write this result (3) under the next coefficient in the row, which is -9. Now, we add the two numbers in that column: -9 + 3 = -6. We write this sum (-6) below the line.

This -6 becomes the next number we work with. We repeat the process: multiply -6 by k = 1, which gives us -6. Write -6 under the next coefficient, which is -8. Add these two numbers: -8 + (-6) = -14. Write -14 below the line. See the pattern? Multiply, add, repeat. We're on a roll! Let's keep going. Multiply -14 by k = 1, which gives us -14. Write -14 under the next coefficient, which is 28. Add these two numbers: 28 + (-14) = 14. Write 14 below the line. One more step to go! Multiply 14 by k = 1, which gives us 14. Write 14 under the last coefficient, which is -14. Add these two numbers: -14 + 14 = 0. Write 0 below the line. And that’s it! We've completed the synthetic division.

1 | 3 -9 -8 28 -14
  |     3 -6 -14 14
  |________________________
  | 3 -6 -14 14  0

Interpreting the Results and Finding p(k)

Okay, we've done the synthetic division, and now we need to understand what those numbers at the bottom mean. The numbers below the line (3, -6, -14, 14, and 0) are super important. The last number on the right (in our case, 0) is the remainder. Remember the Remainder Theorem we talked about earlier? The remainder is the value of p(k). So, p(1) = 0. The other numbers (3, -6, -14, and 14) are the coefficients of the quotient when you divide p(x) by (x - k). In this case, the quotient is 3x³ - 6x² - 14x + 14. However, for our main question, we're primarily interested in the remainder. Since the remainder is 0, this tells us something very important: k = 1 is a zero of the polynomial p(x). This means that if you plug 1 into the polynomial, you'll get 0 as the result.

Think of it like this: because the remainder is 0, (x - 1) is a factor of p(x). In other words, we can write p(x) as (x - 1) times the quotient we found ( 3x³ - 6x² - 14x + 14). This is a powerful result because it helps us factor the polynomial and find its other zeros. If the remainder had been something other than 0, say 5, then we would know that p(1) = 5 and that 1 is not a zero of the polynomial. But in our case, we hit the jackpot! We found a zero. So, to answer the original question: Is k = 1 a zero of the polynomial? The answer is a resounding yes! And, just for good measure, we also found that p(1) = 0. Woohoo!

Conclusion: k=1 is a Zero of p(x)

So, to wrap it all up, we used synthetic division to efficiently determine if k = 1 is a zero of the polynomial p(x) = 3x⁴ - 9x³ - 8x² + 28x - 14. By performing the synthetic division, we found that the remainder is 0, which means that p(1) = 0 and k = 1 is indeed a zero of the polynomial. Synthetic division is a fantastic tool for this kind of problem, saving us a lot of time and effort compared to directly plugging in the value of k or using polynomial long division. Plus, it gives us the added bonus of finding the quotient, which can be helpful for further analysis of the polynomial. Keep practicing these steps, and you’ll become a synthetic division master in no time! Remember, math is like any other skill – the more you practice, the better you get. So, keep at it, and you'll be solving polynomial problems like a pro. Great job, guys! See you next time!