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Functions of polynomials with different degrees

Question 1) Determine whether the function is a polynomial function. If it is, state the degree. If it is not, tell why not.

a) Polynomial of degree 4
b) Polynomial of degree 7

Question 2) Form a polynomial whose real zeros and degree are given. Zeros: -1, 0, 5. Degree: 3
Write a polynomial with integer coefficients and a leading coefficient of 1.
Simplify the answer.

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Solution Preview

1) Yes, 7(x - 5)^2(x^2 + 6) is a polynomial.

Note that "7" is a polynomial (a polynomial that's constant),

(x - 5)^2 is a quadratic polynomial (one that is of degree 2),

and x^2 + 6 is a quadratic polynomial (one that is of degree 2).

The product of finitely many polynomials (in the same variable, such as "x" in this case) is a polynomial in that same variable.

Also, the degree of a polynomial that is formed from the product of finitely many polynomials (all in the same variable) is the sum of the degrees of the individual polynomials in the product.

The degree of "7" is 0 (since it's a constant polynomial).

The degree of (x - 5)^2 is ...

Solution Summary

A complete, detailed solution is provided for each question. The individual pieces of the solutions are used to explain how the overall solutions are obtained.