Chapter 8.3, Problem 49HP

To determine

Summarize the method that can be used to multiply polynomials.

Explanation of Solution

Given:

Method that can be used to multiply polynomials

Here are the steps required for Multiplying Polynomials:

Step 1:	Distribute each term of the first polynomial to every term of the second polynomial. Remember that when you multiply two terms together, must multiply the coefficient (numbers) and add the exponents.
Step 2:	Combine like terms (if you can).

Let’s take an example:

Example 1 − Multiply: $3x^2\left(4x^2 –5x+7\right)$

Step 1: Distribute each term of the first polynomial to every term of the second polynomial. In this case, distribute the $3x^2$ .

  $\Rightarrow 3x^2(4x^2-5x+7)$

Step 2: Combine like terms. In this case, there are no like terms.

  $\begin{array}{l}\Rightarrow 3x^2\times 4x^2-3x^2\times 5x+3x^2\times 7\\ \Rightarrow 12x^4-15x^3+21x^2\end{array}$

Another example,

Example 2 − Multiply: $\left(3x–4y\right)\left(5x–2y\right)$

Step 1: Distribute each term of the first polynomial to every term of the second polynomial. In this case, distribute the $3x$ and the $–4y$ .

  $\begin{array}{l}\Rightarrow \left(3x–4y\right)\left(5x–2y\right)\\ \Rightarrow 3x\times 5x-3x\times 2y-4y\times 5x+4y\times 2y\end{array}$

Step 2: Combine like terms.

  $\Rightarrow 15x^2-26xy+8y^2$

Conclusion:

Therefore, this is the method that can be used to multiply polynomials.