Question

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

Answer 1

Answer:

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

Step-by-step 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!