Is the difference of two polynomials always a polynomial? Explain.

Yes, the difference of two polynomials is always a polynomial. Moreover, any linear combination of two (or more) polynomials is a polynomial.

To prove this, recall the definition of polynomials of one variable. They are finite sums of expressions of the form `a*x^k,` where a is a constant, x is the variable and k is a non-negative integer.

For two polynomials, `P = sum_(k=0)^n a_k x^k` and `Q= sum_(k=0)^m b_k x^k,` its linear combination is

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Yes, the difference of two polynomials is always a polynomial. Moreover, any linear combination of two (or more) polynomials is a polynomial.

To prove this, recall the definition of polynomials of one variable. They are finite sums of expressions of the form `a*x^k,` where a is a constant, x is the variable and k is a non-negative integer.

For two polynomials, `P = sum_(k=0)^n a_k x^k` and `Q= sum_(k=0)^m b_k x^k,` its linear combination is

`R= rP + sQ=sum_(k=0)^(g) (r a_k + s b_k) x^k,`

where g = max(n,m) and zeros are used as the spare coefficients.

We see that R is also a polynomial. The difference is obtained when r = 1 and s = -1.

The same is true for a linear combinations of several polynomials and when they have several variables.