Finding Zeros Of Polynomial Functions If X=3 Is A Zero Of F(x)

In the realm of polynomial functions, identifying zeros is a fundamental task. A zero of a polynomial function f(x) is a value of x for which f(x) = 0. These zeros, also known as roots, provide crucial insights into the behavior and properties of the polynomial. This article delves into the process of finding zeros of polynomial functions, specifically focusing on scenarios where one zero is already known. We will explore techniques such as polynomial division and factoring, empowering you to efficiently uncover additional zeros. Let's consider the problem: if x = 3 is a zero of the polynomial function f(x) = x³ + x² - 17x + 15, find another zero of f(x) by division or factoring. This problem serves as an excellent illustration of how to apply these methods in practice. By mastering these techniques, you will gain a deeper understanding of polynomial functions and their applications in various fields.

Understanding Polynomial Zeros

Before diving into the solution, let's solidify our understanding of polynomial zeros. A polynomial function is an expression consisting of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. The degree of a polynomial is the highest power of the variable in the expression. For instance, the function f(x) = x³ + x² - 17x + 15 is a polynomial function of degree 3.

A zero of a polynomial function f(x) is a value x = a that satisfies the equation f(a) = 0. Graphically, these zeros correspond to the points where the polynomial's graph intersects the x-axis. Zeros can be real or complex numbers, and a polynomial of degree n has at most n zeros (counting multiplicity). The Factor Theorem provides a vital link between zeros and factors of a polynomial. It states that if x = a is a zero of f(x), then (x - a) is a factor of f(x). Conversely, if (x - a) is a factor of f(x), then x = a is a zero of f(x). This theorem is the cornerstone of our approach to finding additional zeros when one zero is already known.

In our example, we are given that x = 3 is a zero of f(x) = x³ + x² - 17x + 15. This implies that (x - 3) is a factor of f(x). Our strategy will involve using this information to divide the polynomial and obtain a quotient, which will help us identify other zeros.

Utilizing Polynomial Division

Polynomial division is a fundamental technique for dividing one polynomial by another. It is particularly useful when we know a factor of the polynomial, as in our case. Since we know that x = 3 is a zero of f(x) = x³ + x² - 17x + 15, we can divide f(x) by (x - 3) to find the remaining factors. The process mirrors long division with numbers. We set up the division as follows:

 x - 3 | x³ + x² - 17x + 15

We start by dividing the leading term of the dividend (x³) by the leading term of the divisor (x), which gives us x². We write x² above the x² term in the dividend. Next, we multiply the divisor (x - 3) by x², which yields x³ - 3x². We subtract this from the dividend:

 x - 3 | x³ + x² - 17x + 15
       - (x³ - 3x²)
       -------------
             4x² - 17x

We bring down the next term (-17x) and repeat the process. We divide the leading term of the new dividend (4x²) by the leading term of the divisor (x), which gives us 4x. We write +4x above the -17x term. Multiply the divisor (x - 3) by 4x, resulting in 4x² - 12x. Subtract this from the current dividend:

 x - 3 | 4x² - 17x
       - (4x² - 12x)
       -------------
            -5x + 15

Bring down the last term (+15). Divide the leading term of the new dividend (-5x) by the leading term of the divisor (x), which gives us -5. Write -5 above the +15 term. Multiply the divisor (x - 3) by -5, resulting in -5x + 15. Subtract this from the current dividend:

 x - 3 | -5x + 15
       - (-5x + 15)
       -------------
             0

The remainder is 0, which confirms that (x - 3) is indeed a factor of f(x). The quotient we obtained is x² + 4x - 5. This means we can write f(x) as:

f(x) = (x - 3)(x² + 4x - 5)

Factoring the Quadratic Quotient

Now that we have expressed f(x) as a product of (x - 3) and the quadratic x² + 4x - 5, we can find the remaining zeros by factoring the quadratic. Factoring a quadratic involves expressing it as a product of two linear factors. To factor x² + 4x - 5, we look for two numbers that multiply to -5 and add to 4. These numbers are 5 and -1. Therefore, we can factor the quadratic as:

x² + 4x - 5 = (x + 5)(x - 1)

Substituting this back into our expression for f(x), we get:

f(x) = (x - 3)(x + 5)(x - 1)

Now we have factored the polynomial completely. To find the zeros, we set each factor equal to zero and solve for x:

x - 3 = 0 => x = 3 x + 5 = 0 => x = -5 x - 1 = 0 => x = 1

Thus, the zeros of f(x) are x = 3, x = -5, and x = 1. We were given that x = 3 is a zero, and we have found two additional zeros: x = -5 and x = 1. Therefore, the correct answer is B. x=1.

Alternative Approach: Factoring by Grouping (If Applicable)

While polynomial division is a robust method, some polynomials can be factored directly using techniques like factoring by grouping. This method involves rearranging terms and factoring out common factors from pairs of terms. However, factoring by grouping is not always applicable, and it may not be straightforward for all polynomials.

In our example, f(x) = x³ + x² - 17x + 15, factoring by grouping is not immediately obvious. Therefore, polynomial division is a more reliable approach in this case. However, it's worth being aware of factoring by grouping as a potential alternative for certain polynomials.

Key Takeaways and Summary

In this article, we explored the process of finding zeros of polynomial functions, focusing on the scenario where one zero is already known. We learned that a zero of a polynomial f(x) is a value of x that makes f(x) = 0. We utilized the Factor Theorem, which states that if x = a is a zero of f(x), then (x - a) is a factor of f(x), and vice versa. We employed polynomial division to divide the given polynomial by the factor corresponding to the known zero. This allowed us to obtain a quotient, which we then factored to find the remaining zeros.

Specifically, we addressed the problem: if x = 3 is a zero of the polynomial function f(x) = x³ + x² - 17x + 15, find another zero of f(x) by division or factoring. We performed polynomial division of f(x) by (x - 3), obtaining the quotient x² + 4x - 5. We then factored this quadratic to (x + 5)(x - 1). This led us to identify the zeros x = -5 and x = 1, in addition to the given zero x = 3. Therefore, another zero of f(x) is x = 1. We also briefly discussed factoring by grouping as an alternative technique, although it was not directly applicable to our example.

By mastering these techniques, you will be well-equipped to find zeros of polynomial functions and gain a deeper understanding of their behavior and properties. This knowledge is essential in various areas of mathematics and its applications.

Practice Problems

To reinforce your understanding, try solving the following practice problems:

1. If x = -2 is a zero of f(x) = x³ + 6x² + 11x + 6, find the other zeros.
2. Given that x = 1 is a zero of f(x) = 2x³ - 3x² - 3x + 2, find the remaining zeros.
3. If x = 4 is a zero of f(x) = x³ - 9x² + 20x - 12, determine the other zeros.

By working through these problems, you will solidify your skills in finding zeros of polynomial functions using division and factoring.