Remainder Theorem Determine Polynomial Zeros F(x)=4x^4+16x^3-16x^2+13x-35
In this article, we will delve into the application of the remainder theorem to ascertain whether a given number, denoted as c, is a zero of a polynomial function. The remainder theorem provides a powerful and efficient method for evaluating polynomial functions at specific values, thereby enabling us to identify potential zeros. Specifically, we will explore the polynomial function $f(x) = 4x^4 + 16x^3 - 16x^2 + 13x - 35$ and investigate whether c = -5 and c = 4 are zeros of this function. Understanding how to use the remainder theorem is crucial for various mathematical applications, including factoring polynomials, finding roots, and simplifying algebraic expressions.
The remainder theorem is a fundamental concept in polynomial algebra that establishes a connection between polynomial division and the value of the polynomial at a specific point. In essence, the theorem states that if a polynomial f(x) is divided by a linear divisor of the form (x - c), the remainder obtained from this division is equal to f(c). This seemingly simple relationship has profound implications for determining whether a number c is a zero of the polynomial. A zero of a polynomial, by definition, is a value of x that makes the polynomial equal to zero. Therefore, if f(c) = 0, then c is a zero of f(x). This theorem provides an efficient way to check for zeros without resorting to more complex methods like factoring or using the quadratic formula. The power of the remainder theorem lies in its ability to quickly evaluate polynomials and identify potential zeros, making it an indispensable tool in polynomial algebra.
Applying the Remainder Theorem to $f(x) = 4x^4 + 16x^3 - 16x^2 + 13x - 35$
In this section, we will apply the remainder theorem to our given polynomial $f(x) = 4x^4 + 16x^3 - 16x^2 + 13x - 35$. Our goal is to determine if the given numbers, c = -5 and c = 4, are zeros of this polynomial. To do this, we will evaluate f(-5) and f(4) using synthetic division, which is an efficient method for dividing a polynomial by a linear factor of the form (x - c). The remainder obtained from the synthetic division will tell us the value of the polynomial at the given c. If the remainder is zero, then c is a zero of the polynomial. This process allows us to systematically check each value and determine whether it satisfies the condition for being a zero. By applying the remainder theorem in this way, we can avoid directly substituting the values into the polynomial, which can be computationally intensive for higher-degree polynomials. Instead, synthetic division simplifies the process and provides a straightforward way to find the remainders and, consequently, identify the zeros.
Part 1: Determining if c = -5 is a Zero
To ascertain whether c = -5 is a zero of the polynomial $f(x) = 4x^4 + 16x^3 - 16x^2 + 13x - 35$, we will employ synthetic division. Synthetic division is a streamlined method for dividing a polynomial by a linear divisor of the form (x - c). In this case, our divisor is (x - (-5)), which simplifies to (x + 5). We set up the synthetic division by writing the coefficients of the polynomial in a row, followed by the value of c (-5) to the left. The process involves bringing down the first coefficient, multiplying it by c, adding the result to the next coefficient, and repeating this process until we reach the end. The last number in the bottom row is the remainder. If the remainder is zero, then c is a zero of the polynomial. For c = -5, we perform the synthetic division as follows:
-5 | 4 16 -16 13 -35
| -20 20 -20 35
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4 -4 4 -7 0
The remainder obtained from the synthetic division is 0. According to the remainder theorem, this means that f(-5) = 0. Therefore, c = -5 is a zero of the polynomial $f(x) = 4x^4 + 16x^3 - 16x^2 + 13x - 35$. This result confirms that when x is replaced with -5 in the polynomial, the expression evaluates to zero, satisfying the definition of a zero of a polynomial.
Part 2: Determining if c = 4 is a Zero
Now, let's investigate whether c = 4 is a zero of the polynomial $f(x) = 4x^4 + 16x^3 - 16x^2 + 13x - 35$. We will again use synthetic division, this time with the divisor (x - 4). We set up the synthetic division similarly to the previous part, using the coefficients of the polynomial and the value c = 4. The synthetic division process involves bringing down the first coefficient, multiplying it by c, adding the result to the next coefficient, and repeating until we reach the end. The remainder will indicate the value of f(4). If the remainder is zero, then c = 4 is a zero of the polynomial. Performing the synthetic division for c = 4, we have:
4 | 4 16 -16 13 -35
| 16 128 448 1844
---------------------------
4 32 112 461 1809
The remainder obtained from this synthetic division is 1809. Since the remainder is not zero, according to the remainder theorem, f(4) ≠0. Therefore, c = 4 is not a zero of the polynomial $f(x) = 4x^4 + 16x^3 - 16x^2 + 13x - 35$. This result indicates that substituting 4 for x in the polynomial does not result in zero, thus confirming that 4 is not a zero of the given polynomial.
In conclusion, we have successfully applied the remainder theorem to determine whether the given numbers c = -5 and c = 4 are zeros of the polynomial $f(x) = 4x^4 + 16x^3 - 16x^2 + 13x - 35$. Through the use of synthetic division, we found that f(-5) = 0, indicating that c = -5 is indeed a zero of the polynomial. Conversely, we found that f(4) = 1809, which is not equal to zero, thereby confirming that c = 4 is not a zero of the polynomial. The remainder theorem provides an efficient and reliable method for identifying zeros of polynomials, and its application is crucial in various areas of mathematics, including algebra, calculus, and numerical analysis. Understanding and utilizing this theorem allows for the simplification of complex polynomial evaluations and contributes to a deeper understanding of polynomial behavior.
By mastering the remainder theorem, one can readily ascertain whether a given number is a zero of a polynomial, which is a fundamental skill in polynomial algebra. The application of synthetic division, as demonstrated, streamlines the process, making it an invaluable tool for students and professionals alike.
