How do you Write a Polynomial Function with Given Zeros

To write a polynomial function with given zeros, we first need to convert the zeroes into factors by expressing each zero as (x - a) where a is the zero. For example, if the zeros are x₁, x₂, . . . ,x_n​, the polynomial function can be written as:

P(x) = k(x − x₁)(x − x₂) . . . (x − x_n)

Where k is a constant. By multiplying these factors together, we can obtain the polynomial function in its standard form.

Zeros (or roots) of a polynomial function are the values of the variable x that make the polynomial equal to zero. In other words, if P(x) is a polynomial function, then the zeros are the solutions to the equation P(x) = 0.

Read More about the Zeros of Polynomials.

Steps to Write a Polynomial Function with Given Zeros

To write polynomials with given zeros, we can use the following steps:

Step 1: Identify the Zeros: Determine the zeros of the polynomial. Let's say the given zeros are a, b, and c.

Step 2: Write Factors for Each Zero: For each zero, a, b, and c, write a corresponding factor of the polynomial. If a is a zero, then (x - a) is a factor. Similarly, (x - b) and (x - c) are factors for zeros b and c, respectively.

Step 3: Form the Polynomial: Multiply the factors to form the polynomial. If the zeros are a, b, and c, the polynomial P(x) can be written as: P(x) = k(x - a)(x - b)(x - c) where k is a non-zero constant (typically k = 1 unless otherwise specified).

Step 4: Expand the Polynomial (Optional): If needed, you can expand the factors to express the polynomial in standard form (a sum of terms).

Solved Example of Polynomial Function with Given Zeros

Suppose you are given the zeros 2, -3, and 4.

Step 1: Identify the Zeros: Zeros are 2, -3, and 4.

Step 2: Write Factors: The factors corresponding to these zeros are: (x - 2), (x + 3), and (x - 4)

Step 3: Form the Polynomial: Multiply the factors to get the polynomial: P(x) = (x - 2)(x + 3)(x - 4)

Step 4: Expand the Polynomial (Optional): Expand the factors to express the polynomial in standard form: P(x) = (x - 2)(x + 3)(x - 4)

First, multiply two of the factors:(x − 2)(x + 3) = x² + 3x − 2x − 6 = x² + x − 6

Now, multiply the result by the third factor:(x² + x − 6)(x − 4) = x³ − 4x² + x² − 4x − 6x + 24 = x³ − 3x² − 10x + 24

So, the polynomial in standard form is: P(x) = x³ − 3x² − 10x + 24

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