Question

What is the product of the polynomial 2x^3 + 8x^2 + x + 3 times the
monomial x?

Answer 1

`\huge\boxed{2x^4 + 8x^3 + x^2 + 3x}`

Explanation:

When we multiply a polynomial by a specific monomial, all terms in the polynomial get multiplied by the monomial.

So we can do this by term.

`2x^3 \cdot x`

Multiplying x by x to the something power will just increase the exponent on the x term by one (since powers are just x multiplied by itself). This means the exponent here will increase by 1, so this becomes
`2x^4`.

Same logic applies to the second term:
`8x^2`, raise the exponent by one,
`8x^3`.

For the third term, if there is no exponent on a term you can assume it's exponent is 1 (so
`x^1` in this case - as
`x^1=x`). Again, using the same logic, this turns into
`x^2`.

For the last term, when we multiply a constant by a variable, we get a term with a variable and a coefficient - the constant becomes the coefficient and the variable stays the variable. Therefore,
`3 \cdot x = 3x`.

Adding these terms all together gets us
`2x^4 + 8x^3 + x^2 + 3x`.

Hope this helped!