No.
Algebra
You can find explanation and examples here: http://en.wikipedia.org/wiki/Polynomial_division
It is a polynomial (monomial). It is a polynomial (monomial). It is a polynomial (monomial). It is a polynomial (monomial).
You can evaluate a polynomial, you can factorise a polynomial, you can solve a polynomial equation. But a polynomial is not a specific question so it cannot be answered.
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Distributive Property
Distributive
Yes, but that is not the only property used.
No. A matrix polynomial is an algebraic expression in which the variable is a matrix. A polynomial matrix is a matrix in which each element is a polynomial.
It is useful to know the linear factors of a polynomial because they give you the zeros of the polynomial. If (x-c) is one of the linear factors of a polynomial, then p(c)=0. Here the notation p(x) is used to denoted a polynomial function at p(c) means the value of that function when evaluated at c. Conversely, if d is a zero of the polynomial, then (x-d) is a factor.
Polynomial vs non polynomial time complexity