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Functions of polynomials with different degrees

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Question 1) Determine whether the function is a polynomial function. If it is, state the degree. If it is not, tell why not.
G(x)=7(x-5)^2(x^2+6)

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

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Algebra .

1) Which of the following is/are TRUE for the graph of EVERY polynomial function?

It must have either a minimum or a maximum value
It must be continuous
It must have a y-intercept
It must have at least one x-intercept

2) Which of the following are odd degree polynomials?

x(x-4)^2-11
x^3+x^2-4
(x^2+2)^5+2x
x^5+x^7+x^8+2
x^3+x-11
x^5+4
x^5+x+1/x

3) Is f(x)=13^x+3x^8-12x+16 a polynomial function?

Is f(x)=12x^17/9+13x-4 a polynomial function?

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