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 x ₁ , x ₂ , . . . ,x _n , the polynomial function can be written as:

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

What is a Polynomial Function?

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form of a polynomial function in one variable xxx is:

$P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0$

a _n , a _n−1 , . . . , a ₁ , a ₀ are constants known as coefficients,
x is the variable,
n is a non-negative integer, representing the degree of the polynomial (the highest power of x).

Zeros of Polynomial Functions

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 Zeros of Polynomials.

Steps to Write a Polynomial Function with Given Zeros

To write polynomial 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).

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) = x 2 + 3x − 2x − 6 = x 2 + x − 6

Now, multiply the result by the third factor:(x 2 + x − 6)(x − 4) = x 3 − 4x 2 + x 2 − 4x − 6x + 24 = x 3 − 3x 2 − 10x + 24

So, the polynomial in standard form is: P(x) = x 3 − 3x 2 − 10x + 24

FAQs

What are zeros of a polynomial function?

Zeros (or roots) of a polynomial function are the values of the variable x that make the polynomial equal to zero. For a polynomial P(x), the zeros are the solutions to the equation P(x) = 0.

How do you determine the factors of a polynomial from its zeros?

Each zero aaa of a polynomial function corresponds to a factor of the form (x – a). To form a polynomial with given zerosx ₁ , x ₂ , . . . ,x _n , you write it as:

P(x) = k(x − x ₁ )(x − x ₂ ) . . . (x − x _n )

where k is a constant (often taken as 1 for simplicity).

What if a zero has a multiplicity greater than 1?

If a zero has a multiplicity greater than 1, it means that the factor corresponding to that zero is repeated. For example, if the zero 3 has a multiplicity of 2, and -1 is another zero, the polynomial function would be: