# Determine whether the following state-ments are true and give an explanation or counterexample. a) All polynomials are rational functions, but not all

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 $f×f$ is a quadratic polynomial.
c) If f and g are polynomials, then the degrees of $f×g$ and $g×f$ are equal.
d) To graph $g\left(x\right)=f\left(x+2\right)$, shift the graph of f 2 units to the right.
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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.
Hence, the given statement "All polynomials are rational functions, but not all rational functions are polynomials" is true statement.
Jeffrey Jordon