Chapter 4.6, Problem 50WE

(a)

To determine

To solve: To multiply the given terms.

Explanation of Solution

1. Given Information:

The given expression is $\left(x+1\right)\left(x^2-x+1\right)$

To multiply the terms in given expression, use distributive law and combine the like terms.

  $\begin{array}{c}\left(x+1\right)\left(x^2-x+1\right)=x\left(x^2-x+1\right)+1\left(x^2-x+1\right)\\ =\left(x^3-x^2+x\right)+\left(x^2-x+1\right)\\ =\left(x^3-x^2+x+x^2-x+1\right)\\ =x^3+1\end{array}$

The product of the given terms is $x^3+1$

2. Given Information:

The given expression is $\left(x+1\right)\left(x^3-x^2+x-1\right)$

To multiply the terms in given expression, use distributive law and combine the like terms.

  $\begin{array}{c}\left(x+1\right)\left(x^3-x^2+x-1\right)=x\left(x^3-x^2+x-1\right)+1\left(x^3-x^2+x-1\right)\\ =\left(x^4-x^3+x^2-x\right)+\left(x^3-x^2+x-1\right)\\ =x^4-x^3+x^2-x+x^3-x^2+x-1\\ =x^4-1\end{array}$

The product of the given terms is $x^4-1$

3. Given Information

The given expression is $\left(x+1\right)\left(x^4-x^3+x^2-x+1\right)$

To multiply the terms in given expression, use distributive law and combine the like terms.

  $\begin{array}{c}\left(x+1\right)\left(x^4-x^3+x^2-x+1\right)=x\left(x^4-x^3+x^2-x+1\right)+1\left(x^4-x^3+x^2-x+1\right)\\ =\left(x^5-x^4+x^3-x^2+x\right)+\left(x^4-x^3+x^2-x+1\right)\\ =x^5-x^4+x^3-x^2+x+x^4-x^3+x^2-x+1\\ =x^5+1\end{array}$

The product of the terms is $x^5+1$

(b)

To determine

To predict the product of the given terms.

Explanation of Solution

4. Given Information:

The given expression is $\left(x+1\right)\left(x^5-x^4+x^3-x^2+x-1\right)$

To multiply the terms in given expression, use distributive law and combine the like terms.

Using the above parts, we can generalise the product. The product is $x^{n+1}-1$ ,where $n$ the degree of the second polynomial is.

  $\begin{array}{c}\left(x+1\right)\left(x^5-x^4+x^3-x^2+x-1\right)=\left(x^{5+1}-1\right)\\ =x^6-1\end{array}$

The product of the given terms is $x^6-1$

5. Given Information:

The given expression is $\left(x+1\right)\left(x^6-x^5+x^4-x^3+x^2-x+1\right)$

To multiply the terms in given expression, use distributive law and combine the like terms.

Using the above parts, we can generalise the product.

The product is $x^{n+1}-1$ ,where $n$ is the odd degree of the second polynomial.

The product is $x^{n+1}+1$ ,where $n$ is the even degree of the second polynomial.

In the given expression the degree of second polynomial is $6$ which is even. Therefore, the product is $x^{n+1}+1$ , where $n$ is the degree of the second polynomial.

  $\begin{array}{c}\left(x+1\right)\left(x^6-x^5+x^4-x^3+x^2-x+1\right)=\left(x^{6+1}+1\right)\\ =x^7+1\end{array}$

The product of the given terms is $x^7+1$