Tazmin Horton

2020-12-09

Determine whether the following state-ments are true and give an explanation or counterexample.

a) All polynomials are rational functions, but not all rational functions are polynomials.

b) If f is a linear polynomial, then$f\times f$ is a quadratic polynomial.

c) If f and g are polynomials, then the degrees of$f\times g$ and $g\times f$ are equal.

d) To graph$g\left(x\right)=f(x+2)$ , shift the graph of f 2 units to the right.

a) All polynomials are rational functions, but not all rational functions are polynomials.

b) If f is a linear polynomial, then

c) If f and g are polynomials, then the degrees of

d) To graph

Ezra Herbert

Skilled2020-12-10Added 99 answers

Step 1

(a) Given:

All polynomials are rational functions, but not all rational functions are polynomials.

Determine whether the following state-ments are true and give an explanation or counterexample.

Step 2

Explanation- A polynomial of degree n has at most n real zeros. The degree of a polynomial function determines the end behavior of its graph. A rational function is a function of the form$f\left(x\right)=P\left(x\right)Q\left(x\right)$ or $f\left(x\right)=\frac{P\left(x\right)}{Q\left(x\right)}$ , where P(x) and Q(x) are both polynomials.

Every polynomial function is a rational with$Q\left(x\right)=4.$ A function that cannot be written in the form of a polynomial, such as , is not a rational function.

So, the given statement "All polynomials are rational functions, but not all rational functions are polynomials" is true statement.

Answer:

Hence, the given statement "All polynomials are rational functions, but not all rational functions are polynomials" is true statement.

(a) Given:

All polynomials are rational functions, but not all rational functions are polynomials.

Determine whether the following state-ments are true and give an explanation or counterexample.

Step 2

Explanation- A polynomial of degree n has at most n real zeros. The degree of a polynomial function determines the end behavior of its graph. A rational function is a function of the form

Every polynomial function is a rational with

So, the given statement "All polynomials are rational functions, but not all rational functions are polynomials" is true statement.

Answer:

Hence, the given statement "All polynomials are rational functions, but not all rational functions are polynomials" is true statement.

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