Math Problem Statement

Find a polynomial function that has the given zeros. (There are many correct answers.)

0, 1, 5

Solution

To find a polynomial function with the given zeros 00, 11, and 55, we can write the polynomial as a product of factors corresponding to these zeros.

The factors will be of the form (xzero)(x - \text{zero}). So, the factors for the given zeros are:

  • x0=xx - 0 = x
  • x1x - 1
  • x5x - 5

Now, multiply these factors together to get the polynomial:

f(x)=x(x1)(x5)f(x) = x(x - 1)(x - 5)

Now expand this expression:

  1. Multiply the first two factors:

x(x1)=x2xx(x - 1) = x^2 - x

  1. Now multiply by the third factor (x5)(x - 5):

(x2x)(x5)=x35x2x2+5x=x36x2+5x(x^2 - x)(x - 5) = x^3 - 5x^2 - x^2 + 5x = x^3 - 6x^2 + 5x

Thus, one possible polynomial function is:

f(x)=x36x2+5xf(x) = x^3 - 6x^2 + 5x

Final Answer:

f(x)=x36x2+5xf(x) = x^3 - 6x^2 + 5x

This is a polynomial with the given zeros 00, 11, and 55. There are infinitely many such polynomials because you can multiply this polynomial by any constant, and the resulting function will still have the same zeros.

Would you like a more detailed explanation of the process? Here are some related questions for further practice:

  1. What happens to the polynomial if we multiply it by a constant, say 2?
  2. How do you determine the degree of the polynomial from the number of zeros?
  3. Can a polynomial with real coefficients have complex zeros?
  4. How would you verify that these zeros satisfy the polynomial?
  5. How do you find a polynomial if given complex or repeated zeros?

Tip: A polynomial of degree nn has at most nn zeros, including multiplicity.

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Math Problem Analysis

Mathematical Concepts

Algebra
Polynomials
Factoring

Formulas

Factoring polynomial from zeros: f(x) = (x - zero1)(x - zero2)...

Theorems

Fundamental Theorem of Algebra

Suitable Grade Level

Grades 10-12