## `multcoeffs` -- multiply the coefficients of a polynomial with a factor

#### Introduction

`multcoeffs(`p, c`)` multiplies all coefficients of the polynomial `p` with the factor `c`.

#### Call(s)

`multcoeffs(p, c)`
`multcoeffs(f, <vars,> c)`

#### Parameters

`p` | - | a polynomial of type `DOM_POLY`
`c` | - | an arithmetical expression or an element of the coefficient ring of `p`
`f` | - | a polynomial expression
`vars` | - | a list of indeterminates of the polynomial: typically, identifiers or indexed identifiers

#### Returns

a polynomial of type `DOM_POLY`, or a polynomial expression, or `FAIL`.

### Overloadable:

`p`, `f`

#### Related Functions

`coeff`, `degree`, `degreevec`, `lcoeff`, `ldegree`, `lterm`, `nterms`, `nthcoeff`, `nthmonomial`, `nthterm`, `poly`, `tcoeff`

#### Details

- A polynomial expression `f` is first converted to a polynomial with the variables given by `vars`. If no variables are given, they are searched for in `f`. See `poly` about details of the conversion. `FAIL` is returned if `f` cannot be converted to a polynomial. After multiplication with `c`, the result is converted to an expression.
- For a polynomial expression `f`, the factor `c` may be any arithmetical expression. For a polynomial `p` of type `DOM_POLY`, the factor `c` must be convertible to an element of the coefficient ring of `p`.
- `multcoeffs` is a function of the system kernel.

#### Example 1

Some simple examples:

    >> multcoeffs(3*x^3 + x^2*y^2 + 2, 5)

                                   3      2  2
                               15 x  + 5 x  y  + 10

    >> multcoeffs(3*x^3 + x^2*y^2 + 2, c)

                                         3      2  2
                              2 c + 3 c x  + c x  y

    >> multcoeffs(poly(x^3 + 2, [x]), sin(y))

                                       3
                          poly(sin(y) x  + 2 sin(y), [x])

#### Example 2

Mathematically, `multcoeffs(`f, c`)` is the same as `f*c`. However, `multcoeffs` produces an expanded form of the product which depends on the indeterminates:

    >> f := 3*x^3 + x^2*y^2 + 2:
       multcoeffs(f, [x], c), multcoeffs(f, [y], c),
       multcoeffs(f, [z], c)

                     3      2  2     2  2         3
          2 c + 3 c x  + c x  y , c x  y  + c (3 x  + 2),

                   3    2  2
             c (3 x  + x  y  + 2)

    >> delete f:

#### Changes

- no changes

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