Chapter 6.3, Problem 9LC

Solution

To find the commons and the differences in the process of multiplying radical expressions and polynomial expressions.

Interpretation:

To multiply radical expressions, the process is to multiply the radicands together while keeping their product under the same radical symbol.

For example:

  $\begin{array}{l}a\sqrt{x}\times b\sqrt{y}=ab\sqrt{x\times y}\\ \Rightarrow a\sqrt{x}\times b\sqrt{y}=ab\sqrt{xy}\end{array}$

To multiply polynomial expressions, the first step is to multiply each term from one polynomial to all terms in the other polynomial using the distributive property. The second step is to add the powers of the same variables using the exponent rule.

  $\begin{array}{l}(a{x}^2+bx+c)\times (dx+e)=dx(a{x}^2+bx+c)+e(a{x}^2+bx+c)\\ \Rightarrow (a{x}^2+bx+c)\times (dx+e)=ad{x}^3+bd{x}^2+cdx+ae{x}^2+bex+ce\\ \Rightarrow (a{x}^2+bx+c)\times (dx+e)=ad{x}^3+(bd+ae)x^2+(cd+be)x+ce\end{array}$

In the process of multiplying radical expressions, the radicals and coefficients are grouped alike as grouping the coefficients and variables in multiplying polynomial expressions.

The difference is that the result of the multiplication of polynomial is always polynomial, while the result of multiplying radical expressions can be a polynomial.