Polynomial Function With Given Zeros: Examples & Solutions

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Polynomial Function with Given Zeros: Examples & Solutions

Let's dive into constructing polynomial functions with the least degree and integral coefficients, given specific zeros. This is a fundamental concept in algebra, and we'll break it down step-by-step with examples. We'll tackle how to build these functions from their roots, considering scenarios with leading coefficients and constant terms. So, grab your pencils, guys, and let's get started!

Understanding Polynomial Functions and Zeros

Before we jump into the examples, let's quickly recap what polynomial functions and their zeros are. A polynomial function is an expression consisting of variables (usually 'x') and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. Think of it like this: f(x) = ax^n + bx^(n-1) + ... + c, where 'a', 'b', and 'c' are coefficients, and 'n' is a non-negative integer (the degree of the polynomial).

The zeros of a polynomial function are the values of 'x' that make the function equal to zero. In other words, they are the x-intercepts of the polynomial's graph. If a number 'r' is a zero of a polynomial function f(x), then (x - r) is a factor of f(x). This is a crucial concept for constructing our polynomials.

The least degree part is also important. It means we want the simplest possible polynomial that satisfies the given conditions. We'll achieve this by including only the factors necessary to create the desired zeros.

Building Blocks: Factors and Roots

At the heart of constructing polynomials from zeros is understanding the relationship between factors and roots. Remember that each zero corresponds to a factor. If r is a zero, then (x - r) is a factor. This principle allows us to work backward from the zeros to the factored form of the polynomial. For instance, if we know that 2 is a zero, we immediately know that (x - 2) is a factor. If -3 is a zero, then (x - (-3)) or (x + 3) is a factor. The goal is to identify all the necessary factors and then multiply them together to obtain the polynomial function.

To ensure integral coefficients (meaning no fractions), we sometimes need to adjust our factors. If we have a zero that's a fraction, like 2/3, directly using (x - 2/3) would lead to fractional coefficients when we expand the polynomial. To avoid this, we can manipulate the factor. In the case of 2/3, we multiply the entire factor by the denominator (3) to get (3x - 2). This gives us the same zero (if you set 3x - 2 = 0, you'll find x = 2/3), but now the coefficients are integers. This technique is essential when dealing with rational zeros.

The Role of Leading Coefficients and Constant Terms

Often, problems specify a desired leading coefficient or a constant term. The leading coefficient is the coefficient of the term with the highest power of 'x' in the polynomial. The constant term is the term without any 'x' (the term that's just a number). These constraints add an extra layer to the problem-solving process.

To achieve a specific leading coefficient, after you've constructed the polynomial from its zeros, you might need to multiply the entire polynomial by a constant. For example, if you build a polynomial with zeros that lead to a leading coefficient of 1, but the problem requires a leading coefficient of 40, you'd simply multiply the whole polynomial by 40. This scaling doesn't change the zeros of the polynomial; it only affects the coefficients.

The constant term is the product of the constant parts of each factor. Setting a specific constant term can influence the overall scaling and require a bit more manipulation of the polynomial. You might need to adjust the constant multiplier to ensure the polynomial has the desired constant term without altering the zeros.

Let's move on to the examples and put these concepts into action!

Example 9: Zeros 3, 2/5, 0, 4/5

Okay, let's tackle our first problem. We're asked to write a polynomial function with the least degree and integral coefficients, given the zeros 3, 2/5, 0, and 4/5. This is where our understanding of factors comes into play. Remember, each zero corresponds to a factor.

  • Zero 3 gives us the factor (x - 3).
  • Zero 2/5 gives us the factor (x - 2/5). But to get integral coefficients, we'll use (5x - 2).
  • Zero 0 gives us the factor x.
  • Zero 4/5 gives us the factor (x - 4/5). Again, for integral coefficients, we'll use (5x - 4).

So, our initial polynomial function, f(x), can be written as:

f(x) = x(x - 3)(5x - 2)(5x - 4)

Now, we need to expand this to get the polynomial in standard form (where terms are arranged in descending order of their exponents). Let's do this step by step:

First, multiply (5x - 2) and (5x - 4):

(5x - 2)(5x - 4) = 25x^2 - 20x - 10x + 8 = 25x^2 - 30x + 8

Now, multiply the result by (x - 3):

(x - 3)(25x^2 - 30x + 8) = 25x^3 - 30x^2 + 8x - 75x^2 + 90x - 24 = 25x^3 - 105x^2 + 98x - 24

Finally, multiply by x:

x(25x^3 - 105x^2 + 98x - 24) = 25x^4 - 105x^3 + 98x^2 - 24x

So, the polynomial function of least degree with integral coefficients that has the zeros 3, 2/5, 0, and 4/5 is:

f(x) = 25x^4 - 105x^3 + 98x^2 - 24x

This is a fourth-degree polynomial (because the highest power of x is 4), which makes sense given that we had four zeros.

Example 10: Zeros 3/4, -1, -4/5 with a Leading Coefficient of 40

Alright, let's move on to the second problem. This time, we have zeros of 3/4, -1, and -4/5, and we need to construct a polynomial with a leading coefficient of 40. This adds a little twist, but we've got this!

Let's start by finding the factors corresponding to each zero, keeping in mind the need for integral coefficients:

  • Zero 3/4 gives us the factor (x - 3/4). To get integral coefficients, we'll use (4x - 3).
  • Zero -1 gives us the factor (x - (-1)), which simplifies to (x + 1).
  • Zero -4/5 gives us the factor (x - (-4/5)). To get integral coefficients, we'll use (5x + 4).

Our initial polynomial function, g(x), looks like this:

g(x) = (4x - 3)(x + 1)(5x + 4)

Now, let's expand this step by step. First, let's multiply (4x - 3) and (x + 1):

(4x - 3)(x + 1) = 4x^2 + 4x - 3x - 3 = 4x^2 + x - 3

Next, multiply the result by (5x + 4):

(4x^2 + x - 3)(5x + 4) = 20x^3 + 16x^2 + 5x^2 + 4x - 15x - 12 = 20x^3 + 21x^2 - 11x - 12

So far, we have the polynomial:

20x^3 + 21x^2 - 11x - 12

Notice that the leading coefficient (the coefficient of the x^3 term) is 20. But we need a leading coefficient of 40! To achieve this, we simply multiply the entire polynomial by 2:

2(20x^3 + 21x^2 - 11x - 12) = 40x^3 + 42x^2 - 22x - 24

Therefore, the polynomial function of least degree with integral coefficients, zeros 3/4, -1, and -4/5, and a leading coefficient of 40 is:

g(x) = 40x^3 + 42x^2 - 22x - 24

This is a third-degree polynomial, as expected with three zeros.

Example 11: Zeros 3, 1/2, -5 with a Constant Term of [Missing Value]

Okay, let's tackle the final problem. Here, we have zeros 3, 1/2, and -5, and we need to find a polynomial with a specific constant term. Unfortunately, the constant term is missing in the original problem statement. Let's assume, for the sake of demonstration, that the constant term should be -15. If you have the actual value, simply replace -15 in the final steps.

First, let's find the factors corresponding to each zero:

  • Zero 3 gives us the factor (x - 3).
  • Zero 1/2 gives us the factor (x - 1/2). For integral coefficients, we'll use (2x - 1).
  • Zero -5 gives us the factor (x + 5).

Our initial polynomial function, h(x), is:

h(x) = (x - 3)(2x - 1)(x + 5)

Let's expand this step by step. First, multiply (x - 3) and (2x - 1):

(x - 3)(2x - 1) = 2x^2 - x - 6x + 3 = 2x^2 - 7x + 3

Now, multiply the result by (x + 5):

(2x^2 - 7x + 3)(x + 5) = 2x^3 + 10x^2 - 7x^2 - 35x + 3x + 15 = 2x^3 + 3x^2 - 32x + 15

So, we have the polynomial:

2x^3 + 3x^2 - 32x + 15

The constant term here is +15. However, we want a constant term of -15. To achieve this, we need to multiply the entire polynomial by -1:

-1(2x^3 + 3x^2 - 32x + 15) = -2x^3 - 3x^2 + 32x - 15

Thus, the polynomial function with zeros 3, 1/2, -5, and a constant term of -15 is:

h(x) = -2x^3 - 3x^2 + 32x - 15

This is a third-degree polynomial, consistent with having three zeros.

Key Takeaways and Tips

  • Zeros to Factors: Remember the fundamental connection: if 'r' is a zero, then (x - r) is a factor. Adjust factors with fractional zeros to ensure integral coefficients.
  • Expanding Polynomials: Take it step-by-step to avoid errors. Multiply pairs of factors first, then multiply the results.
  • Leading Coefficient Adjustment: Multiply the entire polynomial by a constant to achieve the desired leading coefficient.
  • Constant Term Adjustment: Similar to the leading coefficient, you can adjust the constant term by multiplying the entire polynomial by a constant, often -1 to change the sign.
  • Missing Information: If a problem has missing information (like the constant term in our Example 11), clearly state your assumption and proceed with the solution.

By following these steps and understanding the underlying concepts, you can confidently construct polynomial functions from their zeros, even with additional constraints like leading coefficients and constant terms. Keep practicing, and you'll become a pro in no time! Guys, if you have any question just let me know. Thanks!