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Chapter 5: Problem 77

Find each product. In each case, neither factor is a monomial. $$\left(x^{2}+2 x+1\right)\left(x^{2}-x+2\right)$$

Short Answer

Expert verified
\(x^{4} + x^{3} + x^{2} + 3x + 2\)

Step by step solution

01

- Distribute First Term from First Parenthesis

Start by multiplying the first term of the first parenthesis, \(x^{2}\), by every term of the second parenthesis. This gives us \(x^{2} \cdot x^{2} = x^{4}\), \(x^{2} \cdot -x = -x^{3}\), and \(x^{2} \cdot 2 = 2x^{2}\). Thus, we get \(x^{4} - x^{3} + 2x^{2}\).
02

- Distribute Second Term from First Parenthesis

then multiply the second term of the first parenthesis, \(2x\), by every term of the second parenthesis. This gives us \(2x \cdot x^{2} = 2x^{3}\), \(2x \cdot -x = -2x^{2}\), and \(2x \cdot 2 = 4x\). Combined with the result from the previous step we get, \(x^{4} - x^{3} + 2x^{2} + 2x^{3} - 2x^{2} + 4x\).
03

- Distribute Third Term from First Parenthesis

Now, multiply the third term of the first parentheses, \1\, by every term of the second parenthesis. This gives us \(1 \cdot x^{2} = x^{2}\), \(1 \cdot -x = -x\), and \(1 \cdot 2 = 2\). Combined with the result from the previous steps we get, \(x^{4} - x^{3} + 2x^{2} + 2x^{3} - 2x^{2} + 4x + x^{2} - x + 2\).
04

- Combine Like Terms

Once all terms have been distributed, we can combine like terms. This results in \(x^{4} + (2x^{3} - x^{3}) + (2x^{2} - 2x^{2} + x^{2}) + (4x - x) + 2\) which simplifies to \(x^{4} + x^{3} + x^{2} + 3x + 2\).

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Most popular questions from this chapter

Use a vertical format to find each product. $$\begin{array}{r}2 x^{3}+x^{2}+2 x+3 \\\\\quad x+4 \\\\\hline\end{array}$$

Perform the indicated operations. $$4 x^{2}\left(5 x^{3}+3 x-2\right)-5 x^{3}\left(x^{2}-6\right)$$

Find each of the products in parts (a)-(c). a. \((x-1)(x+1)\) b. \((x-1)\left(x^{2}+x+1\right)\) c. \((x-1)\left(x^{3}+x^{2}+x+1\right)\) d. Using the pattern found in parts (a)-(c), find $(x-1)\left(x^{4}+x^{3}+x^{2}+x+1\right) without actually multiplying.

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I'm working with two monomials that I cannot add, although I can multiply them.

Perform the indicated computations. Express answers in scientific notation. $$\frac{\left(1.2 \times 10^{6}\right)\left(8.7 \times 10^{-2}\right)}{\left(2.9 \times 10^{6}\right)\left(3 \times 10^{-3}\right)}$$
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